UNIVERSITY  OF  CALIFORNIA 
AT  LOS  ANGELES 


J3KJVERSITY 


_ 
LIBRARY 


A    TREATISE 


INTEGRAL  CALCULUS 


FOUNDED  ON   THE 


METHOD   OF  RATES 


WILLIAM  WOOLSEY    JOHNSON 

Professor  of  Mathematics    at   the    United   States   Naval  Academy 
A  n  napolis,    Ma  ryland 


FIRST    EDITION 
FIRST   THOUSAND 


NEW   YORK 

JOHN  WILEY    &   SONS 
LONDON:  CHAPMAN  &  HALL,  LIMITED 
1907 


Copyright,  1907 

BY 

WILLIAM   WOOLSEY  JOHNSON 


a  hr  fcrirntifir  $rtEa 

finhrrl  Bntmmnttft  anil  Company 

Netn  fork 


Engineering  * 
Mathematical 

Sciences 

Library  Q  A 


PREFACE. 


THE  present  volume  is  an  enlargement  and  an  extension  of 
my  Elementary  Treatise  on  the  Integral  Calculus,  of  which  a 
revised  edition  was  published  in  1898.  The  enlargement  con- 
sists chiefly  in  a  fuller  treatment  of  formulae  of  reduction,  and  of 
double  and  triple  integrals  in  connection  with  their  geometrical 
representation  by  cylindrical  volumes  and  solids  of  variable 
density  respectively.  The  extension,  which  constitutes  nearly 
half  of  the  volume,  consists  of  Chapter  IV  on  Mean  Values  and 
Probabilities,  Chapter  V  on  Definite  Integrals  including  the 
Eulerian  Integrals,  Fourier's  Series,  etc.,  and  Chapter  VI  on 
Functions  of  the  Complex  Variable. 

The  book  forms  a  companion  volume  to  my  Treatise  on  the 
Differential  Calculus,  founded  on  the  Method  of  Rates  (John  Wiley 
&  Sons,  1904)  to  which  belong  the  references  made  in  the  text. 

The  Integral  Calculus  may  be  said  to  consist  of  two  distinct 
parts:  the  first  concerns  the  reduction  of  integral  expressions  to 
previously  known  functional  forms;  the  second  concerns  the 
mode  of  expressing  a  required  magnitude  in  the  integral  form 
for  subsequent  reduction  when  possible.  While  the  fluxional 
foundation  does  not  affect  the  treatment  of  the  former  (which  is 
essentially  an  inverse  process,  depending  upon  a  body  of  rules 
established  in  the  Differential  Calculus);  in  the  latter,  it  leads 
us  to  regard  the  required  magnitude  as  a  particular  value  of  a 

iii 


iv  PREFACE. 

varying  magnitude,  or  fluent,  of  which  the  fluxion  or  rate  of 
growth  is  to  be  ascertained. 

In  some  of  the  simpler  applications,  the  differential,  or  hypo- 
thetical increment,  which,  in  the  treatment  of  the  subject  here 
followed,  measures  this  rate  of  growth,  admits  of  construction, 
e.g.  in  Arts.  103  and  116.  In  general,  however,  it  is  necessary 
to  regard  the  magnitude  in  question  as  the  limit  of  a  sum.  The 
identification  of  this  limit  with  the  definite  integral  is  effected  in 
Art.  97  by  the  aid  of  the  area  which  has  already  been  shown  to 
represent  the  integral  as  originally  denned.  Accordingly,  the 
integral  expression  for  the  fluent  magnitude  is  for  the  most  part 
found  in  the  usual  manner  by  means  of  its  element.  But,  while 
the  idea  of  the  limit  is  of  course  essential,  the  rate  method  pre- 
sents the  advantage  that  the  integral  is  defined  beforehand,  so 
that  the  student  realizes  that  it  is  the  sum  and  not  the  integral 
which  has  an  approximate  character;  in  other  words,  the  idea 
of  approximation  is  not  carried  into  the  definition  of  the  integral. 

The  direct  representation  of  the  fluent  integral  as  the  ordinate 
of  a  graph,  or  curve  determined  directly  by  the  integral  expression 
itself,  flows  natifrally  from  the  fluxional  point  of  view,  and  is 
developed  at  some  length  in  Arts.  85-91. 

Somewhat  more  than  the  usual  amount  of  space  has  been 
devoted  to  methods  of  integration  and  their  classification  in  such 
manner  that  the  form  of  the  expression  to  be  integrated  shall 
suggest  the  best  method  of  attack.  A  careful  consideration  of 
this  matter  has  led  me  to  lay  particular  stress  upon  trigonometric 
substitutions. 

The  subject  of  Mean  Values  of  continuous  variables  forms  one 
of  the  most  useful  and  characteristic  applications  of  the  Integral 
Calculus.  It  seems  therefore,  next  to  the  strictly  geometrical 
applications,  to  be  entitled  to  ample  treatment,  especially  as  it 
includes  the  important  subjects  of  Centres  of  Gravity  and  Radii 


PREFACE.  V 

of  Gyration,  of  which  the  treatment  is  both  simplified  and 
shortened  by  reference  to  the  underlying  principles  of  Mean 
Values. 

In  the  examples  following  Section  XXI  will  be  found  a  con- 
siderable collection  of  discontinuous  multiple-angle  series,  and  in 
the  text,  Arts.  324-327,  an  interesting  illustration  of  the  relation 
of  such  series  to  the  ordinary  multiple-angle  series,  and  their 
connection  with  the  occurrence  of  divergency. 

To  the  section  on  the  F -function  is  appended  a  logarithmic 
table  of  this  function,  and  also  a  table  of  the  values  of  sn  by  means 
of  which  many  interesting  numerical  computations  may  be  made. 

As  in  the  companion  volume,  the  sections  are  followed  by 
large  collections  of  examples  with  their  answers.  Many  of  the 
results  being  properties  of  well-known  curves  are  rendered  acces- 
sible by  references  inserted  in  the  index. 

W.  WOOLSEY  JOHNSON. 

AUGUST,  1907. 


CONTENTS. 

CHAPTER  L 

ELEMENTARY  METHODS  OF  INTEGRATION. 
I. 

PAGB 

Integrals  ............................................  .  ......  ,....,.,.„..  x 

The  differential  of  a  curvilinear  area  ............................  .  ........  3 

Definite  and  indefinite  integrals  .........................................  4 

Elementary  theorems  ................  .  ..................................  6 

Fundamental  integrals  ..........................................  .  .......  7 

Examples  I  ...............  .  ..............................  ..........  10 

IL 

Direct  integration  ..........  .  .......................................  .  .  .  c  14 

Rational  fractions  ....................................................  ,  15 

Denominators  of  the  second  degree  ......................................  .  16 

Denominators  of  degrees  higher  than  the  second  ..........................  .  19 

Denominators  containing  equal  roots  ....................................  .  21 

General  expression  for  the  numerator  of  a  partial  fraction  .................  ..  22 

Examples  II  .............................................  .  .......  .  ,  26 

III. 

Trigonometric  integrals  ............................  .....  ................  33 

Cases  in  which    sin"*  0  cos"  e  de  is  directly  integrable  .......................  34 

The  integrals     sin"e</0,  and    cosseafe  ....................................  36 

f  ^—  ,     f-*.  ,  and  [*-  .  .                                                 ,  37 
J  sin  e  cos  0     J  sin  e            j  cos  9 


The  integrals 


Vll 


CONTENTS. 


PAGE 

Miscellaneous  trigonometric  integrals 38 

dd 
The  integration  of - 40 

a  +  b  cos  0 

Examples  III 43 


CHAPTER  II. 

METHODS  OF  INTEGRATION  —  CONTINUED. 
IV. 

Integration  by  change  of  independent  variable  ............................  50 

Transformation  of  trigonometric  forms  ..................................  51 

Limits  of  a  transformed  integral  .........................................  53 

The  reciprocal  of  x  employed  as  the  new  independent  variable  ..............  53 

A  power  of  x  employed  as  the  new  independent  variable  ....................  55 

Examples  IV  ....................................................  56 

V, 

Integrals  containing  radicals  ............................................     59 

Radicals  of  the  form  y(ax*  +  i>)  ......................................  .  .     61 

The  integration  of  -  _  -  .  64 

*  ±  d*) 


Transformation  to  trigonometric  forms  ..................................  .  65 

Radicals  of  the  form   y(axy  +  bx  +  c  )  ....................................  67 

The  integrals  f  -  —  -    and     f  -  —  -  ^=  ..............  68 

J  y[(x  -  aX*  -  /?)]  J  V[(x  -  a)  (ft  -x)} 

Exampks  V  .......................................................  70 

VI. 

Integration  by  parts  ........  ...........................................  77 

Geometrical  illustration  ................................................  78 

Applications  ..........................................................  78 

•Formulae  of  reduction  ..................................................  81 


Reduction  of  J  sin"*  6  M  and     cos"1  BdB 82 

Reduction  of  f  sin™  9  cos*  6  dQ 84 


CONTENTS.  ix 


I 


Reduction  of  I  cosm  0  cos  n  0  dd. 


Reduction  of  algebraic  forms 90 

General  method  of  deriving  a  formula  of  reduction 91 

Development  of  an  integral  in  series 93 

Bernoulli's  series 96 

Taylor's  theorem. 95 

Examples  VI 97 

VII. 

The  integral  and  its  limits 105 

Condition  of  continuity 107 

Graph  of  an  integral 1 09 

Multiple-valued  integrals 112 

Formulas  of  reduction  for  definite  integrals 116 

Change  of  independent  variable  in  a  definite  integral 119 

The  integral  regarded  as  the  limit  of  a  sum 121 

Additional  formulae  of  integration 124 

Examples  VII 125 

CHAPTER  III. 

GEOMETRICAL  APPLICATIONS — DOUBLE  AND  TRIPLE  INTEGRALS. 
VIII. 

Areas  generated  by  variable  lines  having  fixed  directions 129 

Application  to  the  witch 130 

Application  to  the  parabola  when  referred  to  oblique  coordinates 132 

The  employment  of  an  auxiliary  variable 132 

Areas  generated  by  rotating  variable  lines 134 

The  area  of  the  lemniscate 135 

The  area  of  the  cissoid 136 

A  transformation  of  the  polar  formula; 136 

Application  to  the  folium 137 

Examples  VIII 140 

IX. 

The  volumes  of  solids  of  revolution 147 

The  volume  of  an  ellipsoid 149 

Solids  of  revolution  regarded  as  generated  by  cylindrical  surfaces 156 

Examples  IX 151 


CONTENTS. 


Double  integrals 155 

Limits  of  the  double  integral 156 

The  area  of  integration 159 

Change  of  the  order  of  integration 160 

Triple  integrals 163 

Integration  over  a  known  volume 1 65 

Representation  of  a  triple  integral  by  a  mass  of  variable  density 1 66 

Examples  X 168 

XI. 

The  polar  element  of  area 172 

Transformation  of  a  double  integral 174 

Cylindrical  coordinates ...  176 

Solids  of  revolution  with  polar  coordinates 178 

Polar  coordinates  in  space 179 

Spherical  coordinates , 181 

Volumes  in  general 182 

Examples  XI 185 

XII. 

Rectification  of  plane  curves 189 

Change  of  sign  of  ds 190 

Polar  coordinates 190 

Rectification  of  curves  of  double  curvature 101 

Rectification  of  the  loxodromic  curve 192 

Examples  XII 193 

XIII. 

Surfaces  of  solids  of  revolution 198 

Quadrature  of  surfaces  in  general i99 

The  determination  of  surfaces  by  polar  coordinates 202 

Examples  XIII 203 

XIV. 

Areas  generated  by  straight  lines  moving  in  planes 206 

Applications 207 

Sign  of  the  generated  area 209 

Areas  generated  by  lines  whose  extremities  describe  closed  circuits 210 

Amsler's  planimeter 211 

Examples  XIV 213 


CONTENTS. 


XV. 

PAGE 

Approximate  expressions  for  areas  and  volumes 215 

Simpson's  rules 217 

Cotes'  method  of  approximation 218 

Weddle's  rule 219 

The  five-eight  rule 219 

The  comparative  accuracy  of  Simpson's  first  and  second  rules 220 

The  application  of  these  rules  to  solids 220 

Woolley's  rule 221 

Examples  XV 222 


CHAPTER  IV. 

MEAN  VALUES  AND  PROBABILITIES. 
XVI. 

The  average  or  arithmetical  mean 224 

The  weighted  mean 225 

The  mean  of  a  continuous  variable •. 225 

The  mean  ordinate.  .    226 

The  mean  of  equally  probable  values 227 

The  mean  of  a  function  of  two  variables 228 

The  mean  of  values  not  equally  probable 230 

The  centre  of  position  of  n  points 232 

The  centre  of  gravity  of  unequal  particles 234 

The  centre  of  gravity  of  a  continuous  body 235 

Average  squared  distances  of  points  from  a  plane 237 

The  moment  of  inertia  and  radius  of  gyration  of  an  area 237 

The  radius  of  gyration  of  a  solid 238 

Examples  XVI 240 

XVII. 

Mean  distances  from  a  fixed  point 243 

Mean  distances  between  two  variable  points 246 

Mean  distances  connected  with  a  sphere 248 

Random  parts  of  a  line  or  number 251 

Random  division  into  n  parts 252 

Mean  area  of  a  triangle  with  random  vertices 256 

Mean  areas  found  by  the  method  of  centroids 258 

Examples  XVII 263 


xii  CONTENTS. 


XVIII. 

PAGE 

The  measure  of  probability 266 

Probabilities  represented  by  areas '. 268 

Local  probability 269 

The  element  of  probability 273 

Curves  of  probability 276 

Mean  values  under  given  laws  of  probability 282 

Probabilities  involving  selected  points 284 

Selected  points  upon  an  area 286 

Random  lines 287 

Probabilities  involving  variable  magnitudes 289 

Examples  XVIII 292 


CHAPTER  V. 

DEFINITE  INTEGRALS. 

XIX. 

Differentiation  of  a  definite  integral 296 

Integration  under  the  integral  sign 299 

Application  to  the  evaluation  of  definite  integrals 301 

Employment  of  double  integrals 303 

Transformation  by  change  of  variable 306 

Substitution  of  a  complex  value  for  a  constant 310 

Examples  XIX 311 

XX. 

Infinite  values  of  the  function  under  the  integral  sign 315 

Cauchy's  general  and  principal  values 317 

Integrals  with  infinite  limits 318 

Integrals  of  certain  rational  fractions 320 

Frullani's  integral 325 

Integrals  obtained  by  expansion 329 

Series  in  sines  and  cosines  of  multiple  angles 332 

Integrals  developed  in  multiple-angle  series 334 

Integrals  involving  the  expression  A+  B  cos  x(A*  >B2) 336 

Examples  XX .' 339 


CONTENTS.  xiii 


XXI. 

PAGE 

Functions  expressed  in  multiple-angle  series 341 

Fourier's  series 342 

The  series  in  multiple-sines 345 

Developments  containing  both  sines  and  cosines 347 

Discontinuity  of  the  Fourier's  sine-series 350 

Geometrical  illustration 352 

Differentiation  of  multiple-angle  series 353 

Integral  of  multiple-angle  series 354 

Series  obtained  by  transformation 360 

Functions  with  arbitrary  discontinuities 361 

Formulae  involving  both  sines  and  cosines.  .  .  , 364 

Examples  XXI 365 

XXII. 

The  Eulerian  integrals 372 

Gauss's  II-function 374 

The  gamma-function 377 

Transformations  of  the  Eulerian  integrals 379 

Relation  between  the  two  Eulerian  integrals 381 

Reduction  of  integrals  to  gamma-functions 382 

Reduction  of  certain  multiple  integrals 386 

The  function  log  F(i  +#),  and  Euler's  constant 389 

The  logarithmic  derivative  of  F(x) 392 

Examples  XXII 396 

Table  of  log  /*(«) 401 

Table  oj  values  oj  sn 403 


CHAPTER  VI. 

FUNCTIONS  OF  THE  COMPLEX  VARIABLE. 
XXIII. 

Complex  values  of  the  derivative 404 

Conformal  representation -. 406 

Conjugate  functions  of  x  and  y 407 

Two-valued  functions 410 

Multiple-valued  functions 412 

Meaning  of  integration  when  the  variable  is  complex 414 

Integration  around  a  closed  contour 416 


XIV  CONTENTS. 


PAGE 

Integration  about  a  pole 417 

Integrals  of  functions  with  poles 419 

Integration  about  a  branch-point 420 

Integrals  involving  radicals 42 1 

The  modulus  of  a  sum 425 

Power  series  in  the  complex  variable 426 

Circle  and  radius  of  convergence 428 

Taylor's  series 429 

One-valued  function  must  admit  of  infinite  value 431 

Examples  XXIII 432 

INDEX 437 


THE 

INTEGRAL    CALCULUS. 


CHAPTER   I. 

ELEMENTARY  METHODS  OF  INTEGRATION. 


I. 

Integrals. 

f.  IN  an  important  class  of  problems,  the  required  quanti- 
ties are  magnitudes  generated  in  given  intervals  of  time  \vith 
rates  given  in  terms  of  the  time  / ;  or  else,  being  assumed  to 
be  so  generated  concurrently  with  some  other  independent 
variable,  have  rates  expressible  in  terms  of  this  independent 
variable  and  its  rate. 

For  example,  the  velocity  of  a  freely  falling  body  is  known 
to  be  expressed  by  the  equation 

v=&, (0 

in  which  t  is  the  number  of  seconds  which  have  elapsed  since 
the  instant  of  rest,  and  g  is  a  constant  which  has  been  deter- 
mined experimentally.  If  s  denotes  the  distance  of  the  body 


2  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  I. 

at  the  time  /,  from  a  fixed  origin  taken  on  the  line  of  motion, 
v  is  the  rate  of  s ;  that  is, 

ds 


_ 
~ 


hence  equation  (i)  is  equivalent  to 

ds  =  gt  dt,  ........     (2) 

which  expresses  the  differential  of  s  in  terms  of  /  and  dt.  Now 
it  is  obvious  that  \gf  is  a  function  of  /  having  a  differential 
equal  to  the  value  of  ds  in  equation  (2)  ;  and,  moreover,  since 
two  functions  which  have  the  same  differential  (and  hence  the 
same  rate)  can  differ  only  by  a  constant,  the  most  general 
expression  for  s  is 

'£     .......     (3) 


in  which  C  denotes  an  undetermined  constant. 

2.  A  variable  thus  determined  from  its  rate  or  differential 
is  called  an  integral,  and  is  denoted  by  prefixing  to  the  given 

differential  expression  the  symbol    ,  which  is  called  the  integral 
sign.*     Thus,  from  equation  (2)  we  have 


=  \gtdt, 


which  therefore  expresses  that  s  is  a  variable  whose  differential 
is  gtdt ;  and  we  have  shown  that 

\gtdt  =  \gP  +  C. 

The  constant  C  is  called  the  constant  of  integration;  its 
occurrence  in  equation  (3)  is  explained  by  the  fact  that  we 
have  not  determined  the  origin  from  which  s  is  to  be  measured. 

*  The  origin  of  this  symbol,  which  is  a  modification  of  the  long  s,  will  be 
explained  hereafter.    See  Art.  100. 


§  I.]        THE' DIFFERENTIAL    OF  A    CURVILINEAR  AREA.  3 

If  we  take  this  origin  at  the  point  occupied  by  the  body  when 
at  rest,  we  shall  have  s  =  o  when  t  —  o,  and  therefore  from 
equation  (3)  C  —  o;  whence  the  equation  becomes  s  —  \gfi. 


The  Differential  of  a  Curvilinear  Area. 

3.  The  area  included  between  a  curve,  whose  equation  is 
given,  the  axis  of  x  and  two  ordinates  affords  an  instance  of 
the  second  case  mentioned  in  the  first  paragraph  of  Art.  I  ; 
namely,  that  in  which  the  rate  of  the  generated  quantity,  al- 
though not  given  in  terms  of  t,  can  be  readily  expressed  by  means 
of  the  assumed  rate  of  some  other 
independent  variable. 

Let  BPD  in  Fig.  I  be  the  curve 
whose  equation  is  supposed  to  be 
given  in  the  form 


Supposing    the    variable    ordinate       o\         \        \    dx 

PR  to  move  from  the  position  AB 

to  the  position   CD,  the  required 

area  ABDC\s  the  final  value  of  the  FIG.  i. 

variable  area  ABPR,  denoted  by 

A,  which  is  generated  by  the  motion  of  the  ordinate.     The  rate 

at  which  the  area  A  is  generated  can  be  expressed  in  terms  of 

the  rate  of  the  independent  variable  x.     The  required  and  the 

assumed  rates  are  denoted,  respectively,  by  -^-  and  —  ;  and,  to 

express  the  former  in  terms  of  the  latter,  it  Is  necessary  to 
express  dA  in  terms  of  dx.  Since  x  is  an  independent  variable, 
we  may  assume  dx  to  be  constant  ;  the  rate  at  which  A  is  gen- 
erated is  then  a  variable  rate,  because  PR  or  y  is  of  variable 
length,  while  R  moves  at  a  constant  rate  along  the  axis  of  x. 
Now  dA  is  the  increment  which  A  would  receive  in  the  time 


A       R 


S       C 


4  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  J. 

dt,  were  the  rate  of  A  to  become  constant  (see  Diff.  Calc., 
Art.  22).  If,  now,  at  the  instant  when  the  ordinate  passes  the 
position  PR  in  the  figure,  its  length  should  become  constant, 
the  rate  of  the  area  would  become  constant,  and  the  increment 
which  would  then  be  received  in  the  time  dt,  namely,  the 
rectangle  PQSR,  represents  dA.  Since  the  base  RS  of  this 
rectangle  is  dx,  we  have 

dA  =.ydx-  f(x]dx (i) 

Hence,  by  the  definition  given  in  Art.  2,  A  is  an  integral,  and 
is  denoted  by 


A  = 


Definite  Integrals. 

4-.  Equation  (2)  expresses  that  A  is  a  function  of  x,  whose 
differential  \sf(x)dx  ;  this  function,  like  that  considered  in  Art. 
2,  involves  an  undetermined  constant.  In  fact,  the  expres- 
sion f(x]dx  is  manifestly  insufficient  to  represent  precisely 

the  area  ABPR,  because  OA,  the  initial  value  of  x,  is  not  indi- 
cated. The  indefinite  character  of  this  expression  is  removed 
by  writing  this  value  as  a  subscript  to  the  integral  sign  ,  thus, 
denoting  the  initial  value  by  a,  we  write 

A  —    (    f(y\f1r  (^ 

•"•  —  \  J  \x)axi \6) 


in  which  the  subscript  is  that  value  of  x  for  which  the  integral 
has  the  value  zero. 

If  we  denote  the  final  value  of  x  (OC  in  the  figure)  by  b,  the 
area  ABDC,  which  is  a  particular  value  of  A,  is  denoted  by 


DEFINITE  INTEGRALS. 


writing   this    value    of  x    at    the    top    of   the   integral   sign, 
thus, 


ABDC  =  \f(x)dx (4) 

«  a 

This  last  expression  is  called  a  definite  integral,  and  a  and 
b  are  called  its  limits.      In  contradistinction,  the  expression 

f(x]dx  is  called  an  indefinite  integral. 

5.  As  an  application  of  the  general  expressions  given  in  the 
last  two  articles,  let  the  given  curve  be  the  parabola 


Equation  (2)  becomes  in  this  case 

A  =  ( 

Now,  since  \x*  is  a  function  whose  differential  is  x*dx,  this 
equation  gives 


A  -    x*dx  -  \x*  +  C,     .....     (i) 

in  which  C  is  undetermined. 

Now  let  us  suppose  the  limiting  ordinates  of  the  required 
area  to  be  those  corresponding  to  x  —  i  and  x  —  3.  The  vari- 
able area  of  which  we  require  a  special  value  is  now  represented 

by  I  x*dx,  which  denotes  that  value  of  the  indefinite  integral 

which  vanishes  when  x  =  i.  If  we  put  x  =  I  in  the  general 
expression  in  equation  (i),  namely  %x*  +  C,  we  have  -^  +  C; 
hence  if  we  subtract  this  quantity  from  the  general  expression, 
we  shall  have  an  expression  which  becomes  zero  when  x  —  i. 
We  thus  obtain 


6  ELEMENTARY  METHODS  OF  INTEGRATION.    [Alt.  5. 

Finally,  putting,  in  this  expression  for  the  variable  area,  x  =  & 
we  have  for  the  required  area 


6,  It  is  evident  that  the  definite  integral  obtained  by  this 
process  is  simply  the  difference  between  the  values  of  the  indefinite 
integral  at  the  upper  and  lower  limits.  This  difference  may  be 
expressed  by  attaching  the  limits  to  the  symbol  ]  affixed  to  the 
value  of  the  indefinite  integral.  Thus  the  process  given  in  the 
preceding  article  is  written  thus, 


J  iv*  =  ± 


The  essential  part  of  this  process  is  the  determination  of 
the  indefinite  integral  or  function  whose  differential  is  equal  to 
the  given  expression.  This  is  called  the  integration  of  the 
given  differential  expression. 

Elementary    Theorems. 

7.  A  constant  factor  may  be  transferred  from  one  side  of  the 
integral  sign  to  the  other.  In  other  words,  if  m  is  a  constant 
and  u  a  function  of  x, 

mudx  =  m    udx. 

Since  each  member  of  this  equation  involves  an  arbitrary 
constant,  the  equation  only  implies  that  the  two  members  have 
the  same  differential.  The  differential  of  an  integral  is  by 
definition  the  quantity  under  the  integral  sign.  Now  the 
second  member  is  the  product  of  a  constant  by  a  variable 

factor  ;  hence  its  differential  ismdl  \udx\,  that  is,  m  u  dx,  which 
is  also  the  differential  of  the  first  member. 


§  T.'J  ELEMENTAR  Y   THEOREMS.  7 

8.  This  theorem  is  useful  not  only  in  removing  constant 
factors  from  under  the  integral  sign,  but  also  in  introducing 
such  factors  when  desired.  Thus,  given  the  integral 

(xndx\ 

recollecting  that 

1)  =  (n  +  i)xndx, 


we  introduce  the  constant  factor  n  +  i  under  the  integral  sign  ; 
thus, 

\xndx  —  —  -  —  f  (n  +  i)xndx  =  —  —  xn+  z  +  C.  > 
J  n  +  ijv  n  +  i 

9.  If  a  differential  expression  be  separated  into  parts,  its  in- 
tegral is  the  sum  of  the  integrals  of  the  several  parts.     That  is, 
if  u,vtw,'''  are  functions  of  x, 

\(u  +  v  +  w  +  •  •  •  -}dx  =  \u  dx  +  \v  dx  +  \w  dx  +  •  •  • 

For,  since  the  differential  of  a  sum  is  the  sum  of  the  differ- 
entials of  the  several  parts,  the  differential  of  the  second  mem- 
ber is  identical  with  that  of  the  first  member,  and  each  member 
involves  an  arbitrary  constant 

Thus,  for  example, 

|  (2  —  Vx)  dx  =  \2dx  —  \xdx—2x  —  \x  +  C, 

the  last  term  being  integrated  by  means  of  the  formula  deduced 
in  "Art.  8. 

Fundamental  Integrals. 

10.  The  integrals  whose  values  are  given  below  are  called 
the  fundamental  integrals.     The  constants  of  integration  are 
generally  omitted  for  convenience. 


8  ELEMENTARY  METHODS  OF  INTEGRATION.   [Art.  IO. 

Formula  (a)  is  given  in  two  forms,  the  first  of  which  is  de- 
rived in  Art.  8,  while  the  second  is  simply  the  result  of  putting 
n  =  —  m.  It  is  to  be  noticed  that  this  formula  gives  an  indeter- 
minate result  when  n  =  —  i  ;  but  in  this  case,  formula  (&)  may 
be  employed.* 

The  remaining  formulae  are  derived  directly  from  the  for- 
mulae for  differentiation;  except  that  (/'),  (£'),  (I'},  and  (in'} 

*£ 

are  derived  from  (/),  (/£),  (/),  and  (m)  by  substituting  —  for  x. 

c* 

{  n  ,     _  xn  +  I  [dx  _    __  i  ,  . 

)*        -JTTi  )x™-    ~  (m  -  i)  *--'  * 


(A 


— 
log  a 

=  sin0  .............    (d) 

=  -  cos  0.    ...    ........     (e) 

ft 

*  Applying  formula  (a)  to  the  definite  integral    xndx,  we  have 


}a  n+  I 

which  takes  the  form  -  when  n  —  —  I  ;  but,  evaluating  in  the  usual  manner, 


3«+i_fl*-M-,  ^  +  Ilog^-^  +  Ilogfl-| 

=  log  b  -  log  a  ; 
»  +  i       J»  =  — i  i  J«  =  — i 

a  result  identical  with  that  obtained  by  employing  formula  (3). 

f  That  sign  is  to  be  employed  which  makes  the  logarithm  real.    See  Diff.  Calc., 
Art.  do. 


§  I.] 


FUNDAMENTAL  INTEGRALS. 


f 


=  sec  d 


— .  0  „  •  =    cosec  6  cot#  dd  =  —  cosec  0 (i) 

J    sin2  8         } 

r       dx 

\-7-f -gt  =  sin"1  x  • 

J  V(i  -  ^) 


=  ~  cos 


C'. 


(k) 


+  JT      a 


-cot-1-+  C'.    . 
a          a 


C=  - 


C' 


L-^=isec-1-+  C=  -  -  cosec-1  -  +  C  . 
—  a2)       a  a  a  a 


dx 


—  vers 


. 
'1 


dx 


=  vers    - 


.    . 
(m) 

f    f. 
(m) 


IO  ELEMENTARY  METHODS  OF  INTEGRATION.       [Ex.  I. 

Examples  I. 

Find  the  values  of  the  following  integrals  : 

dx 

i. 


'•  I$- 

f  dx 

5- 

J  i  xs 


dx 

' 


X 


- 
3 


5. 

J  o 

6.  f    (.r-i)«^, 
J' 

a_ 

fT  ,  x.  ,  .  ^V~IT      «3 

7.  («  —  &c)V#,  «s^  —  a^  4  =—,- 
Jo                                                                                       3   Jo       3<? 

fa  7/2sjt2  jtr4"!" 

8.  (a  +  xYdx,  asx  +  ^-  +  ax3  +  =  4a\ 

J  -a  2  4_l-o 

faV^c 
9-  Ji  —  »  «  2  log  a. 

10.  -,  log  (  -  *)       =  log  2. 

J-x  *  J-j 


I.]  EXAMPLES.  1 1 


r  (a  +  x)  .  . ,  2 .  o      ,  2\n 

11.        ; — —ax*  2  vx(a  +  %ax  +  ±x  )  |  =  2T.\\  •  a  . 

Ja  Vx  _U 

f 

12.  I  exdx,  e?  —  i. 

J  o 

13.  sin  8d0t  i  —  cos  6. 

^  o 

14.  cos^^cr,  sin^r     =  o. 

Jo  Jo 

15       " — TTI,  tan  0  |    —  i. 

Jo    COS    U 


17 


18 


r    f* 

JtJf    V(^  -   i 


J 


{*"  dx  A-l^         7t 

J6  .  ,  ., ?T  .  sm-1  -      =-r. 

Jo   V  (a   —  x*y  a  Jo       6 


19    If  a  body  is  projected  vertically  upward,  its  velocity  after  /  units 
of  time  is  expressed  by 

v  =  a  —  gt, 

a  denoting  the  initial  velocity  ;  find  the  space  ^i  described  in  the  time 
d  and  the  greatest  height  to  which  the  body  will  rise. 


C/j 
l  =  \  v  dt  =  a/i  —  I^A2, 

Jo 


,  a  a 

when  z>=o,/=—  ,s  =  —  . 

g  2g 


12  ELEMENTARY  METHODS  OF  INTEGRATION.    [Ex.  L 

20.  If  the  velocity  of  a  pendulum  is  expressed  by 


Tit 

v  =  a  cos  — 

2T 


the  position  corresponding  to  /  —  o  being  taken  as  origin,  find  an  ex- 
pression for  its  position  s  at  the  time  /,  and  the  extreme  positive  and 
negative  values  of  s. 


2Ta  Ttt 

s  =  -  sin  —  . 

7t  2T 


s  =  ±  -  when  /  =  T,  37-,  $r,  etc. 

21.  Find  the  area  included  between  the  axis  of  x  and  a  branch  of 
the  curve 

y  =  sin  x.  2. 

22.  Show  that  the  area  between  the  axis  of  x,  the  parabola 

/  = 


and  any  ordinate  is  two  thirds  of  the  rectangle  whose  sides  are  the 
ordinate  and  the  corresponding  abscissa. 

23.  Find  (a)  the  area  included  by  the  axes,  the  curve 


and  the  ordinate  corresponding  to  x  =  i,  and  (/?)  the  whole  area  be- 
tween the  curve  and  axes  on  the  left  of  the  axis  of  y. 


24.  Find  the  area  between  the  parabola  of  the  «th  degree, 

an~ly  =  x", 

and  the  coordinates  of  the  point  (a,  a). 


§  L]  EXAMPLES.  13 

25.  Show   that   the   area  between   the   axis  of  x,  the  rectangular 
hyperbola 


the  ordinate  corresponding  to  x  =  i,  and  any  other  ordinate  is 
equivalent  to  the  Napierian  logarithm  of  the  abscissa  of  the  latter 
ordinate. 

For  this   reason   Napierian   logarithms  are   often   called  hyperbolic 
logarithms. 

26.  Find  the  whole  area  between  the  axes,  the  curve 


and  the  ordinate  for  x  =  a,  m  and  n  being  positive. 

If  n  >  tn, 

if  n  5  m, 
27.  If  the  ordinate  BR  of  any  point  B  on  the  circle 


be  produced  so  that  BR  •  RP  =  a1,  prove  that  the  whole  area  between 
the  locus  of  P  and  its  asymptotes  is  double  the  area  of  the  circle. 

28.  Find  the  whole  area  between  the  axis  of  x  and  the  curve 


29.  Find  the  area  between  the  axis  of  x  and  one  branch  of  the  com- 
panion to  the  cycloid,  the  equations  of  which  are 

x  —  aty  y  =  a  (i  —  cos  ^>). 


14  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  II. 

II. 

Direct  Integration. 

II.  In  any  one  of  the  formulae  of  Art.  10,  we  may  of  course 
substitute  for  x  and  dx  any  function  of  x  and  its  differential. 
For  instance,  if  in  formula  (b)  we  put  x  —  a  in  place  of  x,  we 
have 

f     dx 

-  =  log  (x  —  a)         or         log  (a  —  x), 
]  x      a 

according  as  x  is  greater  or  less  than  a. 

When  a  given  integral  is  obviously  the  result  of  such  a  sub- 
stitution in  one  of  the  fundamental  integrals,  or  can  be  made 
to  take  this  form  by  the  introduction  of  a  constant  factor,  it  is 

said  to  be  directly  integrable.     Thus,     sinmx  dx  is  directly  in- 

tegrable  by  formula  (e) ;  for,  if  in  this  formula  we  put  mx  for  0, 
we  have 

sin  m  x  •  m  dx  =  —  cos  m  x , 
hence 

sin  mx  dx  =  —     sin  m  x  •  m  dx  = cos  m  x . 

j  m  J  m 

So  also  in  J  v(a  +  bx*}  x  dx , 

the  quantity  x  dx  becomes  the  differential  of  the  binomial 
(a  +  bx*}  when  we  introduce  the  constant  factor  2#,  hence  this 
integral  can  be  converted  into  the  result  obtained  by  putting 

(a  +  bx*)  in  place  of  x  in  \y  xdx,  which  is  a  case  of  formula  (a). 
Thus 

2bx  dx  =  — r  (a  • 
3^ 


§11.]  DIRECT  INTEGRATION.  1 5 

12.  A  simple  algebraic  or  trigonometric  transformation 
sometimes  suffices  to' render  an  expression  directly  integrable, 
or  to  separate  it  into  directly  integrable  parts.  Thus,  since 
—  sin  x  dx  is  the  differential  of  cos  x,  we  have  by  formula  (&) 

f  f  sin  x  dx 

tan  x  dx  —         —  —  log  cos  x  . 

J     cos  x 


jtan2  6  dO  =  ((sec2  6-  i}dO  =  tan  6  -  B  ; 
by  (e)  and  (a), 

[sin3  BdB  =\(i-  cos2  0)  sin  Ode  =  -  cos  0  +  j  cos3  d  ; 
by  (/)  and  (a), 


x 


=  sn'  *  - 


Rational  Fractions. 

13.  When  the  coefficient  of  d^  in  an  integral  is  a  fraction 
whose  terms  are  rational  functions  of  x,  the  integral  may  gen- 
erally be  separated  into  parts  directly  integrable.  If  the  de- 
nominator is  of  the  first  degree,  we  proceed  as  in  the  following 
example. 

fj£  _   ^j-    _|_    2 

Given  the  integral        -  -  dx\ 

}      2.X  +  I 

by  division, 

£2_-_£_+J  _  £  _  3.  +  15  _  L_ 
2X  +   I  2        4          4  2.T  +  I* 


1  6              ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  13 

hence 

1  f               3  f 

15  f      dk 

J      2X  +   I 

2j  *            4]  a 
=  ~j  ~  7  +  ¥  loi 

4  J  2.T  +  I 
J  (2X  +  i). 

3      \                                / 

When  the  denominator  is  of  higher  degree,  it  is  evident  that 
we  may,  by  division,  make  the  integration  depend  upon  that  of 
a  fraction  in  which  the  degree  of  the  numerator  is  lower  than 
that  of  the  denominator  by  at  least  a  unit.  We  shall  consider 
therefore  fractions  of  this  form  only. 

Denominators  of  the  Second  Degree. 

14.  If  the  denominator  is  of  the  second  degree,  it  will  (after 
removing  a  constant,  if  necessary)  either  be  the  square  of  an 
expression  of  the  first  degree,  or  else  such  a  square  increased 
or  diminished  by  a  constant.  As  an  example  of  the  first  case, 
let  us,  take 


The  fraction  may  be  decomposed  thus : 

X  +    I  X  —  I    +  2  I  2 


(x  -  I)2  "      (x  -  i)2     ~  x  -  I  "*"  (x  -  I)2 ' 
hence 


[   x  +  i    j    _(    dx  (      dx 

J(F=ip*  -J^i4  'JjF^Ty 

=  log  (x-  \)- 

f  ff   .\_    t 

15.  The  integral 


§  II.J  RATIONAL  FRACTIONS.  1 7 

affords  an  example  of  the  second  case,  for  the  denominator 
may  be  written  in  the  form 

x*  +  2x  +  6  =  (x  +  i)2  +  5. 
Decomposing  the  fraction  as  in  the  preceding  article, 

x  +  3  x  +  i  2 

(x  +  i)2  +  5  ~  (x  +  i)2  +  5  +  (x  +  i)2  +  5 ' 

whence 


)x*  +  2x  +  6'       J(*+i)2  +  5        J(^+i)2  +  5* 

The  first  of  the  integrals  in  the  second  member  is  directly 
integrable  by  formula  (<£),  since  the  differential  of  the  denom- 
inator is  2  (x  +  i)  dx,  and  the  second  is  a  case  of  formula  (£'). 
Therefore 

X  +  3  ,  .   ,         ,    „  -.  2  .X+l 

•  2x  +  o)  -\ — —  tan"1  — : — . 


16.  To  illustrate  the  third  case,  let  us  take 

f    2x  +  i       , 
-3— — ^5^ 

J  "*  "*    ~        ^ 


in  which  the  denominator  is  equivalent  to  (x  —  £)2  —  6^,  and 
can  therefore  be  resolved  into  real  factors  of  the  first  degree. 
We  can  then  decompose  the  fraction  into  fractions  having  these 
factors  for  denominators.  Thus,  in  the  present  example,  as- 
sume 

(0 


X*  —  X  —  6       X—  3        ;tr  +  .2 ' 

in  which  yi  and  B  are  numerical  quantities  to  be  determined. 
Multiplying  by  (x  —  3)  (x  +  2), 

2x  +  i  =  A  (x  +  2)  +  B(x  —  3) (2) 

£-  f  ^ 
:   iLk     r    /** 


1 8  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  1 6. 

Since  equation  (2)  is  an  algebraic  identity,  we  may  in  it  assign 
any  value  we  choose  to  x.     Putting  x  =  3,  we  find 

7  =  5^4,  whence  A  =  -J, 

putting  x  =  —  2, 

—  3  =  —  5^,        whence  ^  =  |. 

Substituting  these  values  in  (i), 

2*  +  i  7  3 

x*-x-6~  5(*-3)      5(*  +  2)' 
whence 

~  3) 


17.  If  the  denominator,  in  a  case  of  the  kind  last  considered, 
is  denoted  by  (x  —  a)  (x  —  b},  a  and  b  are  evidently  the  roots  of 
the  equation  formed  by  putting  this  denominator  equal  to  zero. 
The  cases  considered  in  Art.  14  and  Art.  15  are  respectively 
those  in  which  the  roots  of  this  equation  are  equal,  and  those 
in  which  the  roots  are  imaginary.  When  the  roots  are  real  and 
unequal,  if  the  numerator  does  not  contain  x,  the  integral  can 
be  reduced  to  the  form 

f  dx 

and  by  the  method  given  in  the  preceding  article  we  find 

-. r-7 r.  = -.    log  (x  —  a)  —  log  (x  —  b) 

}(x-a)(x  —  b)      a  —  b\_  'J 

=  — J-zlog^?, '    (A)* 


*  The  formulae  of  this  series  are  collected  together  at  the  end  of  Chapter  II. 
for  convenience  of  reference.     See  Art.  101. 


§11.]           DENOMINATORS  OF   THE   SECOND  DEGREE.  19 

in  which,  when  x  <  a,  log  (a  —  x)  should  be  written  in  place  of 
log  (x  —  a).     [See  note  on  formula  (b),  Art.  10.] 
If  b  —  —  a,  this  formula  becomes 

f     dx          \.x-a  .  ... 

=  —  log  -           (A1) 

^f>          °    v   _L    n  >.          ' 


J  x*  —  a*      20.      &  x  +  a 

Integrals  of  the  special  forms  given  in  (A)  and  (A')  may  be 
evaluated  by  the  direct  application  of  these  formulas.  Thus, 
given  the  integral 

f          dx 

j  2x*  +  3^-  —  2  ' 

if  we  place  the  denominator  equal  to  zero,  we  have  the  roots 
a  =  £,  b  =  —  2;  whence  by  formula  (A}, 

_  i       I    .       x  —  $ 

~       '  g  5 


--i)  (X  +  2)~2 

or,  since  log  (2x  —  i)  differs  from  log  (x  —  |)  only  by  a  con- 
stant, we  may  write 

f          dx  i  ,       2x  —  i 


—  2       5 


Denominators  of  Higher  Degree. 

18.  When  the  denominator  is  of  a  .degree  higher  than  the 
second,  we  may  in  like  manner  suppose  it  resolved  into  factors 
corresponding  to  the  roots  of  the  equation  formed  by  placing  it 
equal  to  zero.  The  fraction  (of  which  we  suppose  the  numerator 
to  be  lower  in  degree  than  the  denominator)  may  now  be  decom- 
posed into  partial  fractions.  If  the  roots  are  all  real  and  un- 
equal, we  assume  these  partial  fractions  as  in  Art.  16 ;  there 
being  one  assumed  fraction  for  each  factor. 

If,  however,  a  pair  of  imaginary  roots  occurs,  the  factor  cor- 


2O  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  1 8. 

responding  to  the  pair  is  of  the  form  (x  —  a)z  +  ft*,  and  the 
partial  fraction  must  be  assumed  in  the  form 

Ax  +  B 


(x  -  of  +  f? ' 

for  we  are  only  entitled  to  assume  that  the  numerator  of  each 
partial  fraction  is  lower  in  degree  than  its  denominator  (other- 
wise the  given  fraction  which  is  the  sum  of  the  partial  fractions 
would  not  have  this  property).  For  example,  given 


Assume 

(^  +  \)(x  -  i)  ~  *»  +  i  +  J^~i'    •    •    •'    CO 
whence 

.».       I        •}    /  ~    T  W  /4  -*•    _l_     P\    _1_    I  -V-2       I        T  \  /"" 

*  ~r  3  —  v*  —  ij  ^./i.*  -f-  Jj)  -f-  ^^t    •(•  \  )L,. 

Since  in  an  identical  equation  the  coefficients  of  the  several 
powers  of  x  must  separately  vanish,  the  coefficients  of  x*,  x 
and  x°  give,  for  the  determination  of  A,  B  and  C,  the  three 
equations 

From  these  we  obtain  A  =  —  2,  B  =  —  i,  C  =  2 ;  hence, 
substituting  in  equation  (i), 

X  +  3 2  2X  +   \ 

therefore 

*  +  3  f    dkr  f  2*  dx        (     dx 


=  2  log  (*—i)_  log(^  +  i)  —  tan"1  jr. 

19.  The  method  of  determining  the  assumed  coefficients 
illustrated  above  makes  it  evident  that,  the  denominator  being 


§  II.]  MULTIPLE  ROOTS.  21 

of  the  nth  degree,  we  must  assume  n  of  these  coefficients  • 
because  we  have  to  satisfy  n  equations,  derived  from  the 
powers  of  x  from  xn  ~ 1  to  x°. 

It  is  evident  that  we  may  take  for  the  denominators  of  the 
partial  fractions  any  two  or  more  factors  of  the  given  denomi- 
nator which  have  no  common  factor  between  themselves,  pro- 
vided we  assume  for  each  numerator  a  polynomial  of  degree 
just  inferior  to  that  of  the  denominator.  But,  since  our  present 
object  is  to  separate  the  given  fraction  into  directly  integrable 
parts,  when  a  squared  factor  such  as  {x  —  of  occurs,  instead 
of  assuming  the  corresponding  partial  fraction  in  the  form 

—2  (which  would  require  to  be  further  decomposed  as  in 
(x  —  a) 

Art.  14),  we  at  once  assume  a  pair  of  fractions  of  the  form 

A  B 


x  -  a   '  (x  —  a? ' 

20.  We  proceed  in  like  manner  when  a  higher  power  of  a 
linear  factor  occurs-.     For  example,  given 


JrJlflwF+l)^*5 


we  assume 


x  +  2  _/4_  £  C_           D    m 

(x  -  i)\x  +  I)  ~  (x  -  if  +  (x  -  \J  +  x  -  i  +  x  +  i1 

whence 

X  +  2 - [^  +  B(x- 1)  +  a^- 1)2](*  +  o  +  ^(* -  o3-  (0 

Putting  x—  i,  we  have  3  =  2A  .'.        A  =  %. 

Putting  ^=  —  i,  we  have        i  =  —  8Z>        .'.        D  =  —  £. 

The  most  convenient  way  to  determine  the  other  coefficients 
is  to  equate  to  zero  the  coefficient  of  x3,  and  to  put  x  —  o.  We 
thus  obtain 


22  ELEMENTARY  METHODS  OF  INTEGRA  TION.    [Art.  2O. 

o  =  C  +  D,     and     2=  A  —  B  +  C—  D, 
from  which  C  =  •§-,  and  B  =  —  £.     Therefore 
f        x  +  2  3  f     dx  i  f     dx          \  [  dx        if  <&r 

1    ___  yV  -y»    -      .    I ,     I    L       _    .    I  . .     _ 

\  Q  /                             \  W--4-  "-      '  ,  O  "T"       r~       \ 

I    /   A- T   \O/   ^y-     |       j  \  O     I  /   >* T    I"                 /ill    'V T   1*  ^1  -»* T                ?s    I 

J    I  «^-             1  1    \*^  ~\       *•  J  ^   J  \"              *  )  *T    J  \  **'  "  '  '    A   I  O    J  **•    "  '  "  1                O  J  ^ 

4 los 


_ 

4(x  -  if      4(>  -  i)       8         x+i 

21.  The  fraction  corresponding  to  a  simple  factor  of  the 
given  denominator  may  be  found,  independently  of  the  other 
partial  fractions,  by  means  of  the  expression  derived  below. 
Denote  the  given  denominator  by  (f>(x),  and  let  it  contain  the 
simple  factor  x  —  a,  so  that 

(x),   ......     (i) 


in  which  ty(x)  does  not  vanish  when  x  =  a. 

Let  f(x]  denote  the  numerator  (it  is  not  necessary  here  to 
suppose  this  to  be  lower  than  (f>(x}  in  degree),  and  let  Q  denote 
the  entire  part  of  the  quotient.  Then  we  may  assume 


<f>(x]  "  (x  -  d)$(x)  r  x  -  a  ^  ^r/ 

in   which  Q  and  P  are   in  general  polynomials.     Clearing  of 
fractions, 

/(*)  =  Q(*  -  aW(*>  +  A<P(*}  +  P(*  -  a). 

Putting  x  =  a,  we  have  (since  neither  P  nor  Q  can  become  in- 
finite) 

f[a)  =  At(a),     whence     A  =         ,  .     .     .     .     (2) 


an  expression  for  the  numerator  A. 

Again,  differentiating  equation  (i),  we  have 


"•]  PARTIAL  FRACTIONS.  2$ 

whence,  putting  x  =  a,  <t>'(a)  =  $(a\  the  substitution  of  which 
in  equation  (2)  gives  another  expression  for  A,  namely, 

A        f(a} 

~~    /.// — r •   •     «     •     •     •     •     •     •     (3) 

As  an  example,  let  us  find  the  value  of 

f  *°  +  2 

I     i   ,    — ^ 5 dx. 

J  X*  +  2JT  —  X*  —  2X 

The  denominator  is  the  product  x(x*  —  i)(x  +  2),  and  the  first 
term  of  the  quotient  is  obviously  x ;  hence  we  assume 

x5  +  2  A         B  C  D 

X*  +  2X*  —  X*  —  2.X  X          X  -\-    I         X  —  \         x  +  2' 

The  coefficient  of  x*  in  the  equation  cleared  of  fractions  gives 
a  =  —  2.  Now  forming  the  fraction 

f(x]    _  X5  +  2 

<p'(x}  ~~  4-r3  -f  6X2  —  2x  —  2' 

we  may  determine  A,  B,  C  and  D  in  accordance  with  expres- 
sion (3)  by  putting  therein  for  x  successively  o,  —  I,  I  and  —  2. 
Thus  A  =  —  i,  B  =  %,  C  =  %,  D  =  $;  whence 

X5  +  2  II  15 

2 +  — r  +  -; v  +      • 


X*  +   2X3  —  X*  —  2X  X          2(X  +  l)         2(x—l) 

and 

f  X5  +  2  X*  (X  +   2)5  4/(>2  —   I) 

-dx  = 2x  +  log 


J  X*  +  2X3  —  X*  —  2X  "2  X 

22.  We  have  seen  that  the  decomposition  of  a  given  frac- 
tion into  partial  fractions  presupposes  a  knowledge  of  the  roots 
of  the  equation  resulting  from  equating  the  given  denominator 
to  zero.  In  the  case  of  the  denominator  xn  —  i,  we  can  employ 
the  expressions  for  the  imaginary  roots  involving  the  circular 
functions  of  certain  angles.  (Diff.  Calc.,  Art.  231).  In  some 
simple  cases  the  factors  are  expressible  in  ordinary  surds.  For 
example,  we  have 


24  ELEMENTARY  METHODS  OF  INTEGRA  T1ON.  [Art.  22. 

^8  _  !  —  (x  _  i)(x  _j_  !)(A-2  +  !)^2  _  x  v/2  +  j^  +  x  y2  +  ^ 

It  is  to  be  noticed  that  when  the  fraction  is  a  rational  func- 
tion of  some  higher  power  of  x,  we  may  simplify  the  process 
of  decomposition.  Thus  it  is  legitimate  to  assume 

x4  A  B 

A-8    y  ~A     -r  ,j*4        I          y  ' 

Jt,        -—      i  J,        -  1  *        -f     I 

because,  if  z  =  x*,  the  numerator  and  factors  of  the  denominator 
are  rational  functions  of  z.  The  first  term  could  be  treated  in 
like  manner  with  respect  to  x2;  but  not  the  second,  since  the 
factors  of.*4  +  i  involve  ;tras  well  as  xz, 

23.  We  have  seen  in  the  preceding  articles  that  the  partial 
fractions  corresponding  to  the  real  roots,  whether  single  or 
multiple,  are  directly  integrable,  and  also  those  corresponding 
to  unrepeated  imaginary  roots.  In  the  following  example  a 
case  of  multiple  imaginary  roots  occurs:  given 


(a -*)(<*  + 
It  is  readily  shown  that  the  partial  fraction  corresponding 

to  a  _  x  is  — — _    and   the    remainder    is    conveniently 

2a3(a  —x) 

found  by  subtracting  this  from  the  given  fraction ;  thus 

a  -f-  x  i  a4  -f-  2a*x  —  2#2#2  —  x* 

(a  -  x)(c?  +  ^)2  ~  2a3(a  -x)=      2a\a  -  x)(tf  +  x*)*    ' 

The  numerator  of  this  remainder  is  found  to  be  divisible  by 
a  —  x,  thus  verifying  the  process ;  and  the  given  integral  re- 
duces to 

I  i    fa3  +  -$a*x  +  ax2  +  x3 

s  l°g  (a  —  x)  -\ »    7-2 2T2 **• 

20?  '       2a*  J  (a*  +  x*)* 

The  integration  of  the  last  term  may  be  effected  by  a  trans- 
formation given  later.  See  Art.  41  and  Art.  76. 

24.  Instead  of  assuming  the  partial  fractions  with  undeter- 


§11.]  RATIONAL  FRACTIONS.  2$ 

mined  numerators,  it  is  sometimes  possible  to  proceed  more 
expeditiously  as  in  the  following  examples: 
Given 

dx\ 


(i+**)' 
putting  the  numerator  in  the  form  i  +  x*  —  x*,  we  have 


dx 

a- 


Treating  the  last  integral  in  like  manner, 

xdx 


log  (i  +  *«)  =  -        +  log 


Again,  given 


putting  the  numerator  in  the  form  (i  +  xf  —  2x  —  x*,  we  have 

i  -,    _  (dx  2  + 

a'    -- 


dx         f       dx        ~  f      dx 


Hence  by  equation  (A),  Art.  17, 

dx  i  x 


26  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  IL 

Examples  II. 

I.  —  log  (a  —  x\ 

}a  —  x* 

dx  i 


'• 


xdx  i 

3'  <  log  (a   + 


.    (*(*- 

Jo 


f    . 

.       (« 

Jo 


7. 


II  \  ** 

x-  id  ~r  mx\  — 

8.    \  (a  +  mxY  dx,  S 


3/0 

COt  2-^ 


sin  2.*  2 

I  —  COS*  X 


10.  cos3  x  sin  x  dx, 

Jo 

f  cos  Q  dQ  i 

11.  cosec  9. 

J    sm  Q  2 

f       ,  sec*  3*  — 

12.  sec  3  x  tan 7~ 

Jo 


EXAMPLES. 


f 

*       I  si^.-% /j'y 

\.  J#     ax, 


.     tan3 

7T 

.      4sec4 

J  o 


•n 

.  M 


24.     sin  (a  —  26)  d$t 


13.    \a-ax,  mloga 


14. 
j 

( i  +  3  sin2jc)3 


15.  (i  +  3  sin1.*)"  sin  X  cos  x  dx, 

r*.  (^  _  ^)  ^  ^T-  o 

16.  -77 —       — sv,  y^aa«      ^ ; 
J  o  V(2ax  —  x  ) 


2 


17.  Jjcos'**,  -  3- 

1 8.  j  sec4  6^/6,  tane  +^tan3Q. 

1 9.  |  tan-  x  dx,  \  t^2  x  +  log  cos  x. 


3 

-sec  x          —, 

4  Jo        4 


'sr  jg  .    \  ^ 

4.  «     '  a 


II  —  log  2 
cot"  e  dfe,  ~2 


-\x 

x)  +  a  vers     — , 
a 


Q'  —  29) 


28  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 

f  cos  x  dx  i  .,..,. 

25'    \a-b^X>  —  logO^sm*). 


f4    dx 

•  t  i  log  2. 

)„  tan*  » 


26 

6 


27-    I  '  *  *  •»  ^Iog2. 

J  „  tan  * ' 

4 
i  I 

i°  2      ^Jjc  J2 

— ; ,  log  (—log*)       =  —  log  2. 

Ji  *  log*  ' Ji 


28.    , 

)  i  X  log  * 
4 


+  e--^' 


tan" 


i  x*  dx  i 

3a    I  ^    .    T  >  -  tan-J*\ 

•  3 

x  dx  i    .      .** 

31-    \~77T* ^n"»  -sin"1-^. 

1  4/(tf  —  * )'  2  a 


dx  i 

32-    I-^T: TI5\»  -^Tsm' 


5-   f     ,  *  ,,  -T-  tan'1 

J  2  +  5*  ' 


I 
i/io 


</*  ^ 

JT'          I          Vl/lnv*    t  1     * 


dx  2     .       _,2*  +   I"!1 

35-        ^t.    ^   .    ,t  T7,tan     '    ~ 


1= 


§  II.]  EXAMPLES.  29 


COS". 


f  V(x*  -  <?}   .    f      f      o?  -  a9        .   "I 

37.          ~  -  '<#?       =    -  77-5  -  jrdfr      , 

J  x  L     J-xVC*   —  a  ')       J' 

-/(*»-.«')-0sec-J-. 

<3f 

38-  (  'T^^  aa(iog2-f). 

J  o  #  ~~  •* 

I**  J+*  3  ^>  4* 


39 


+  ^r  4-  I  ,    i       /    »  \  2  ,2X— 

4°-          _^>    ^  +  log  (^  -^  +  i)  +  -tan 


. 


((2X  +    ^  dX 

-  J      2^+3      >  ^  -  *  +  2  log  (2*  +  3). 

(V 

.    U— :     — 4r<3^,  -log  (2*  +  i) — . 

}(2X  +    I)3  2  2X  +    I 


—  I 


—  2ax  cos  a  + 


i  i-*  —  0  cos  ana        7i  —  a 

: tan      '  : =   : 

a  sm  a.  a  sin  a  2a  sm  a 

— 10 


3°  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  IL 

—  a  sec  a  —  a  tan  a 

47 


f dx  i  -r 

J  -*a  —  2a.r  sec  a  +  a?  '        2#  tan  a         .* 

48. 


2ax  sec  or  +  aa'  2#  tan  a      °  x  —  a  sec  «  +  a  tan  a; ' 

if  4/2               2JT   —    2    —    3  |/2 

^f  —  7*  12               2X  —  2    +    3  1/2* 

•*"<£: 


f  x*dx 

49-  i=3F- 


51 


3-*-  i       ^  £ 


•|(ar+a)^+3)"  2l°g 


',2. 


>•   J  x'  -  x*  -  x  +  i 

J   X  I 


+   2          X  +  3 

x  dx 


^[tan-^  +  log^^'- 


'togfLUL+J^tan-i 

6      &A:+  i          3 


—  X  +   2     ,  9.       X  +  1         I  .        X  —  2 

— 1         dx,  -log h  -  log — — — . 


55-   '  '-""**•' 


4        jc  —  i 


X. 


(      dx  i  ,         (jc  +  i)s  i  _,  2^;  —  i 

57-      — T — *»  7^°S~rn T *"  ~Zr~tan        ~tf * 

f  oc doc  ii  i 

58.       7 rrTI v,  -log(*—  i) log(#S+l) -j— — i. 

J  (x  -  i)   (x 8  +  i)'  2  4  2(^—1) 


§  II.]  EXAMPLES. 


dx 
59 


x  +  xy  +  x') ' 
log  x log  (i  +  x} log  ( i  +  x*) tan  - '  x. 


I  I 

6*      '       6log*  +  - 


f      x*  —  i 
60.    L-TT  —  —.  —  axt 

)X    +  X*   +  I 
(    X*   +  X  —   I 

6l"   J  *»  +  *»-  6 

f  X*dx  I  .         X  —  2 

62.    \-  --  5  -  ,  -log-      -  + 

}x  —  x—  12'  7     &  x  +  2 

f      ^cVjc 

3<  \  (x*  -  i)*' 


1  ,     x*  —  x  +  i 
-log-jT-r      -  — 

2  &  X  •     +  X  +   I 


i        x  —  i 

4  og^T7 


f  2-v"  —  -to*  .  5  A:       i  . 

64-       —  i  -  ^-dx,  ^-tan-1  ---  log 

}    x   —  a  20,  a       40 

,       f 
65         * 


x  —  a 


. 
20,  a       40        x  +  a 

x  dx  i        x*  —  2 


f  dx  i       f  x  +  b  b        _  x~~\ 

00.          ..a      --  ^r-j  -  ;  —  rr  ,        75  —  ;  -  5      lOg       ./    a     ,  -  JT  +  ~  tan     '-      . 

J  (x   +  a)  (x  +  b)  '      b    +  a*  L       V(x   +  »  )      «  ^J 


dx 
7' 


68 

' 


f     x  +  i  x 

09.     — ; ^rdx,  tan-1  x  +  log— 77 IT 

jx(i  +  jr)  &  V(i  +^s) 

f        dx  i         i 

7°-       « /  3  . — c ,  tan ~*x  -\ -5 , 

I*  V*  T  ij  >^      3 


32  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 

dx  X  i 


f          d*  _Ll 

>   Jar  (a  +  &*«)'  ^  10g 


( 

^?r  i          b        a  + 


73> 


74.  Find  the  whole  area  enclosed  by  both  loops  of  the  curve 


75.  Find  the  area  enclosed  between  the  asymptote  corresponding 
to  x  =  a,  and  the  curve 


76.  Find  the  whole  area  enclosed  by  the  curve 

77.  Find  the  area  enclosed  by  the  catenary 

|—    X  X~\ 

the  axes  and  any  ordinate. 

78.  Find  the  whole  area  between  the  witch 
and  its  asymptote.     See  Ex.  23. 


§IIL]  TRIGONOMETRIC  INTEGRALS.  33 


III. 

Trigonometric  Integrals. 

25.    The    transformation,     tan20  —  sec2  6  —  i,    suffices    to 
separate  all  integrals  of  the  form 


[tan*  040, (i) 


in  which  n  is  an  integer,  into  directly  integrable  parts.     Thus, 
for  example, 

[tan5  OdB  =  [tan3  6  (sec2  0-  i)  dQ 
tan40 


-1 


tan3  0  dd. 


4 
Transforming  the  last  integral  in  like  manner,  we  have 

f       ,  -  . .      tan4  0      tan2  8      f 
[tan*0<f0  =  — — +  |tan0</0; 

J  4  2         J 

hence  (see  Art.  12) 

f  tan4  (9      tan20      . 

tan5  9dB  =  - —  log  cos  6. 

j  4.2 

When  the  value  of  n  in  (i)  is  even,  the  value  of  the  final  inte- 
gral will  be  0.     When  n  is  negative,  the  integral  takes  the  form 

f  cot"  0  d0, 
which  may  be  treated  in  a  similar  manner. 


34  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  26. 

26.  Integrals  of  the  form 

[see"  Odd  ........    (2) 

are  readily  evaluated  when  n  is  an  even  number,  thus 
(sec6  Od9  =  [(tan2  0  +  i)2  sec2 


=  [tan4  dsz<?6dd+2  [tan2  0  sec2  0</0  +  [sec3  0  </0 


tan5  0      2  tan3  0 

—  -  —  H  --  +  tan  0. 


If  «  in  expression  (2)  is  odd,  the  method  to  be  explained  in 
Section  VI  is  required. 

Integrals  of  the  form     cosecw  6dO  are  treated  in  like  manner. 

Cases  in  which  sin"*  0  cos"  0  dd  is  directly  integrable. 

27.  If  «  is  a.  positive  odd  number,  an  integral  of  the  form 

[sin"  0  cos*  QdO   .......    (3) 

is  directly  integrable  in  terms  of  sin  6.     Thus, 

[sin2  e  cos5  8d0=  [sin2  0  (i  -  sin2  0)2cos  6dB 


sin3  0      2  sin5  0      sin7  B 
-j-       —        —j- 


This  method  is  evidently  applicable  even  when  m  is  frac- 
tional or  negative.     Thus,  putting  y  for  sin  0, 


lll.j  TRIGONOMETRIC  INTEGRALS.  35 


f cos'  V  f(i  -y)0y  _  f    _, 

.  3  -  «C7  =     5 —  \y  *  ay  — 

Jsin*0  J        y*  K 

hence 

f  cos3  6       _    _      -i_£,f-    _£     3  +  sin2  6 
Jsina  0  33        V(sin  6}  ' 

When  m  in  expression  (3)  is  a  positive  odd  number,  the  in- 
tegral is  evaluated  in  a  similar  manner. 

28.  An  integral  of  the  form  (3)  is  also  directly  integrable 
when  m  +  n  is  an  even  negative  integer,  in  other  words,  when  it 
can  be  written  in  the  form 


J 


in  which  q  is  positive. 
For  example, 

^    ^r2 
sin«  6  cos*  6 


r 
Jsi 

=  f(tan  0)~*  (tan2  0  +  i)  sec2  0</0; 


hence 


_  2       i  2 

~t: 


tan**' 


It  may  be  more  convenient  to  express  the  integral  in  terms 
of  cot  0  and  cosec  0,  thus 

r  =  !cot4  e  (cot8  ^  +  x)  cosec2  ^^ 


cot7  (9      cot5  0 


3^  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  28. 

Integrals  of  the  forms  treated  in  Art.  25  and  Art.  26  are  in- 
cluded in  the  general  form  (3),  Art.  27.  Except  in  the  cases 
already  considered,  and  in  the  special  cases  given  below,  the 
method  of  reduction  given  in  Section  VI  is  required  in  the 
evaluation  of  integrals  of  this  form. 

The  Integrals  sin2  0  dd,  and  |  cos2  0  dd. 

29.  These  integrals  are  readily  evaluated  by  means  of  the 
transformations 

sin2  0  =  £(i  —  cos  20),        and       cos2  0  =  £(i  +  cos  20). 
Thus 

[sin2  BdB  =  %(dd  -  £ [cos  26 dd  =  £0  —  isin  20, 

or,  since  sin  20  =  2  sin  0  cos  0, 

(sin2  0  dO  =  \(0  -  sin  0  cos  0) (B) 

In  like  manner 


sin0cos0) 


Since  sin2  0  +  cos2  0  =  I,  the  sum  of  these  integrals  is  L/0;  ac- 

cordingly we  find  the  sum  of  their  values  to  be  0. 

In  the  applications  of  the  Integral  Calculus,  these  integrals 
frequently  occur  with  the  limits  o  and  \n  ;  from  (B)  and  (C) 
we  derive 


§111.]  TRIGONOMETRIC  INTEGRALS.  37 


r        do          f  dO  ,  f  do 

The  Integrals  \-r-~        —  ,    -  —  -  ,  and          -  . 
J  sin  6  cos  0   Jsm<9  Jcostf 


30.  We  have 


Again,  using  the  transformation, 

sin  0  =  2  sin  \Q  cos  £0, 
we  have 


tan  $V 
hence 

' (£) 


This  integral  may  also  be  evaluated  thus, 
dO       fsm6>^      f  sinOdO 


Since  sin  6dO=  -  ^/(cos  6^,  the  value  of  the  last  integral  is,  by 
formula^'),  Art.  17, 

i        i  -  cos  0      .        /i  -  cos  0 


.        /i 
logr  i 


and,  multiplying  both  terms  of  the  fraction  by  I  —  cos  0,  we 
have 

O  i  —  cos  0 


38  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  31. 

31.  Since  cos  0  =  sin  (£TT  +  0),  we  derive  from  formula  (£), 

f    dO        (        dO  rn      0~\ 

—3  =  -^—TI  --  -a  =  log  tan    -  +  -    .     .     .    (F\ 
}  cos<?      }sm(^7t  +  0)  j_4      2J 


By  employing  a  process  similar  to  that  used  in  deriving  for- 
mula (.£'),  we  have  also 

dB  i  +  sintf 


Miscellaneous   Trigonometric  Integrals. 

32.  A  trigonometric  integral  may  sometimes  be  reduced, 
by  means  of  the  formulae  for  trigonometric  transformation,  to 
one  of  the  forms  integrated  in  the  preceding  articles.  For 
example,  let  us  take  the  integral 

f  dO 

}a  sin  B  -f  b  cos  0' 

Putting  a  =  &  cos  a,  b  =  k  sin  a,    .     .     .     .     (i) 

we  have 

f  d6  __  i_  f       d8 

J  a  sin  B  +  b  cos  B  ~  k.  }  sin  (6  +  a)' 

Hence  by  formula  (E) 


—  :  —  7,  -  1  --  B       I 

J  a  sin  0  +  £  cos  0      k 
or,  since  equations  (i)  give 


i 

~      lo     tan  ~ 


r  ^~i 

a] • 


§  III.]    MISCELLANEOUS   TRIGONOMETRIC  INTEGRALS.  39 

33.  The  expression  sin  md  sin  nO  dd  may  be  integrated  by 
means  of  the  formula 

cos  (m  —  n)  6  —  cos  (m  +  n)  6  =  2  sin  md  sin  nd  ; 
whence 

f  •       a   -      a  JA      s'm(m  —  n)0     sm(m  +  n)6  . 

\smm8smn0  ad  =  —  7  -  (  ----  -r—         —  .     .    (i) 
J  2  (m  —  n)           2  (m  +  n) 

In  like  manner,  from 

cos  (m  —  n)  6  +  cos  (m  +  n)  6  =  2  cos  m6  cos  nd, 
we  derive 

f  a  ja      sin  (»*—»)  0  ,  sin  (m  +  n)Q          .  , 

cos  *#0  cos  n6d6  =  --  >—    —  f  --  1  ---  )  -  •  —  f—  .    .    (2) 
J  2(m  —  n)  2  (m  +  n) 

When  m  =  n,  the  first  term  of  the  second  member  of  each 
of  these  equations  takes  an  indeterminate  form.  Evaluating 
this  term,  we  have 

sin2«#  ,  N 

.....    (3) 


and  cos2^  =  g  +  Sin2.  (4) 


Using  the  limits  o  and  n  we  have,  from  (i)  and  (2),  when  m 
and  n  #?r  unequal  integers, 

fir  ,n 

sin  mQs'm  nOdd  =  \    cos  w0  cos  nB  dd  =  o  ;   .    .    (5) 

•  o  Jo 

but,  when  m  and  n  are  ^##/  integers,  we  have  from  (3)  and  (4) 

=  \\os*n8d6  =-  .....    (6) 

Jo  2 


34.  To  integrate  4/(i  +  cos  ^)  ^/(9,  we  use  the  formula 
2  cos2  \Q  —  i  +  cos  0. 


40  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  34. 

whence  V(i  +  cos  6)  —  ±  4/2  cos  £0, 

in  which  the  positive  sign  is  to  be  taken,  provided  the  value  of 
0  is  between  o  and  n.     Supposing  this  to  be  the  case,  we  have 

[V(i  +  cos  0) dB  =  V2  (cos^Odd 

=  24/2  sin  £0. 
For  example,  we  have  the  definite  integral 

IT 

2  4/(i  +  cos  6}d8  =  2  4/2  sin  -  =  2. 
Jo  4 


Integration  of 7—  — 5. 

y   a  +  b  cos  0 

35.  By  means  of  the  formulae. 

i  =  cos2£0  +  sin2£0       and       cos  0  =  cos2£0  —  sin2£0, 

we  have 

f       dB         _  f dB         

\a  +  b  cos  0  ~  J  (a  +  b)  cos2  ±B  +  (a  -  b}  sin2 10* 

Multiplying  numerator  and  denominator  by  sec2^0,  this  be- 
comes 

f  sec2  \6dQ 

J  a  +  b  +  (a  —  b}  tan2  £0 ' 

and,  putting  for  abbreviation 

tan  £0  =  y, 
we  have,  since  £  sec2^0</0  =  a^, 


__ 


di  +  b  cos  0  tf  +  b  +  (a  —  b)y 


§  III.]   MISCELLANEOUS    TRIGONOMETRIC  INTEGRALS.          41 

The  form  of  this  integral  depends  upon  the  relative  values 
of  a  and  b.  Assuming  a  to  be  positive,  if  b,  which  may  be 
either  positive  or  negative,  is  numerically  less  than  a,  we  may 
put 

a  +  b  _  » 

a  —  b 

The  integral  may  then  be  written  in  the  form 

2      f     dy 


the  value  of  which  is,  by  formula  (£'), 


V 

tan  -  '  ±-  . 


c  (a  -  b) 


Hence,  substituting  their  values  for  y  and  c,  we  have,  in  this 
case, 


If,  on  the  other  hand,  b  is  numerically  greater  than  a,  this 
expression  for  the  integral  involves  imaginary  quantities ;  but 
putting 

b  +  a  _    , 

T  £> 

the  integral  becomes 

2      f     dy 

~j I  ~o  a  » 

b  —  a]  c  —  jr 

the  value  of  which  is,  by  formula  (A1),  Art.  17, 

i        ,      c  +  y 

—r-. r    lOg . 

c(b-a)     *c-y 


42  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  35. 

Therefore,  in  this  case, 

dB  i  V(b+a}+  V(b-a)  tan  \B 

g  -        -  ' 


+  £  cos  6       V  (£2  -  a2)         i/(£  +  tf)-  V(£-«)  tan 

36.  If  ^  <  i,  formula  (£)  of  the  preceding  article  gives 

f        dB  2  .r    /i  -  e.      ,  ."I 

—z,  =  -77  --  a?  tan  MA/  -7     -tanl(?    .    .   (i) 
J  i  +  ^  cos  6       V  (i  —  f)  \_V  i  +  e         -    J 

Putting 

='tant0=tant*f     .....     (2) 


and  noticing  that  0  =  o  when  0  =  o,  we  may  write 

f       _*?  _  -  _  ^_ 
Joi  +*>cos0~4/(i-^)' 


Now,  if  in  equation  (i)  we  put  ^  for  ^  and  change  the  sign  of 
f,  we  obtain 


f  _ 

J0  i  — 

hence,  by  equation  (2), 

f      _^_ 
oi-ecos<f> 


Equations  (3)  and  (4)  are  equivalent  to 
dB  dj> 


i+ecos0 


(5) 


^  dB  fi) 

and 1  =  -77 ^  »      .....     (oj 


§  III.]  TRIGONOMETRIC  INTEGRALS.  43 

the  product  of  which  gives 

(i  +  e  cos  ff)  (i  —  e  cos  <f>)  =  i  —  e*  .     .     .     .     (7) 
By  means  of  these  relations  any  expression  of  the  form 

dB 


1       f 
J(i 


+  e  cos 


where  n  is  a  positive  integer,  may  be  reduced  to  an  integrable 
form.     For 

f          dB  __  f        dO  __  i  m 

J(i  +  e  cos  6)*  ~  Ji  +  e  cos0  (i  +  e  cos  d}"-*  ' 

hence,  by  equations  (5)  and  (7), 


~~  e  cos 


By  expanding  (i  —  e  cos  ^)*~I,  the  last  expression  is  reduced 
to  a  series  of  integrals  involving  powers  of  cos  <f> ;  these  may 
be  evaluated  by  the  methods  given  in  this  section  and  Section 
VI,  and  the  results  expressed  in  terms  of  6  by  means  of  equa- 
tion (2)  or  of  equation  (7). 

Examples  III. 

f      4  tan3  mx      tan  mx 

}  yn  m 

•a 

2.       ta.ri'xdx,  A~il°g2- 

J  o 

_«  /c   i    ,,,\  7/-  •  tan  \u  •+•  oc)  /,         \ 


44  ELEMENTARY  METHODS  OF  INTEGRATION.    [Ex.  IIL 

4.  I    sin3  mx  dx,  yn ' 

Jo 

f  .  ,  sin8©      sin5 9 

5.  sin  0  cos  0^/0,  . 

J  3  5 

6.  f  t/(sin  0)  cos5  0  4/9.  -  sin'0  —  -  sin*  0  +  —  sin^"  9. 
J                                                  3  7  ii 

7.  2  cos4  0  sin'  04/0,  — . 
J°                                                                                        35 

f  sin3  0  4/0  &  A 

J  V(cos0)'  |  cos*  0  -  2  cos«  9. 

9.     -T-J 7— ,        Multiply  by  sin8  0  +  cos"  0.        tan  0  —  cot  9. 

J  sin  6  cos  0 

f     •      3    „ 

>.       —^—dx, 
Jcos  ^ 


10.     J"\ ~  //jc.  See  Art.  2%, 


4 


ii 


f  4/9 

—  r—  ,  4  (tan7  0  -  cot'  0)  +  2  log  tan  9. 

J  sm3  0  cos3  0  ' 

fy(sin  0)</9 
J 


—  s 
cos8  0 


ivrfxdx 


5  cos  JP      3  cos  x 

tan5*  .   tan3* 
3 


f  sin2.*  dx  , .       tan  x  , 

I4'   J    cos'*    '  5 

15.    [sin"  0  cos"  0  </0,  -j^  [26  —  sin  20  cos  20]. 


§  III.]  EXAMPLES.  45 


7t 

2m 


1 6.  I     sirfmxdx, 

f  sin2  0  d6  FTT       8~1 

17.  — ,  log  tan     -  +  -     —  sin  0. 
J     cos0  L4       2J 


(log  3  -  i). 


r       i*  i  .         re  .  an 

. ',  — r—  logtan     -  +  -    . 

Jsm0  +  cos  e»,  V2  L2        8j 

f       dx 

.  , 

J  i  +  cos  x 


21. — ,  i—  cot$x 

J^  i  —  cos  JT 

2    . 

f       dx 

22.  : , 

J  i  ±  sin  x 
Multiply  both  terms  of  the  fraction  by  i  T  sin  x.  tan  x  T  sec  x. 

23-      TT-, ^>  logtan     -  +  -      ±  log  cos  0. 

Jsec0  ±  tan0'  L4       2j 

24.  cos  0  cos  30  </9.     61?^  Art.  33.  f  sin  40  +  %  sin  20. 

7T 

25.  cos  0  cos  20  dQ. 

Jo  3 

7T  1" 

26.  4  sin1 0  sin  20  d8,  $  sin4  6  |    —  i- 

J  o  — 10 

7T 

fa      .  2 

27.  sin  30  sm  20^5,  -• 

•o  J 


46  ELEMENTARY  METHODS  OF  INTEGRATION.     [Ex.  IIL 


28.       sin  m&  cos  nO  dQ, 

•  o 


i  —  cos  (m  +  n)  Q       i  —  cos  (m  —  n)  0 
2  (m  +  n)  2  (m  —  n) 


29.  cos  #  cos  2*  cos  $x  dJr, 

Reduce  products  to  sums  by  means  of  equation  (2),  Art.  33. 

i  Fsin  6*  ,   sin  AJC      sin  2* 
—    —  7  --  1  --    -  H  ---  f-  #    . 
4L     6  4  2  J 

»» 

30.  V  (i  —  COS^)^C,  2  V2. 

Jo 

f  dx  i  .     _,p  ^        "I 

31.  ~i  -  ;  -  ,.,    .  2  —  ,  Ttan       -tan*    . 
Ja2  cos'*  -h  F  sm  x                                    ab  \_a  \ 


( 
'   Ji 


dx  i  j  tan  x 

"  t£        "- 


dx  i  a  +  ^  tan  jf 


cos  A:  — 


f  sin  x  dx 

34.    —n—  •  «  \t  cosMicos*!. 

J  V  (3  cos  x  +  4  sin  #)  ' 

fsin  *  cos8  x  dx 
35<   J  i  +  a"  cos'*  ' 

f    /^ 
Putting  y  /(?r  cos  *,  /^  integral  becomes  —    —  —  —  r«' 


cos*       tan"1  (a  cos  *) 

2         i  Ta 


o  Hi  1 


EXAMPLES.  47 


f 
3  '    J  a 


+  b  sin  9 
Put  sin  9  =  cos  (9  -  \n\  and  use  formulas  (G)  and  (G'). 

2         ,  r  /a  —  b  *  26  ~  **"] 
t£   "T" 


i 
If  a  <  *>  ~  '   10g 


+  a)  +  V  (t  -  a)  tan  ($Q-± 


-  a'  (*  +  «)-   V(^- 


9  - 


_ 

5  +  3  cos  9' 


_ 

5  —  4  cos  9 


f 
'    J     — 

f 
"   J 


( 

4i-  J  ^ 


[ 
J(i 


f- 
>  J    (i 


_    _ 
44>       (i  +*cos9)3 


37'  '  '   2-taniO 

,8     f 
'    J 


tan-M3 


^9  J_        i  +  4^3 


_  log 

8 


acose-i 

d*  tan 


- 
(i  +  e  cose)2' 

i  g  +  cos9  g  sine 


i+<?cosfl       i  — 


48  ELEMENTARY  METHODS  OF  INTEGRATION,  [Ex.  III. 


f  p  cos  x  +  a  sin  x  , 

45. -~-. dx, 

J     J  a  cos  x  +  b  sm  x 


Solution : — 

By  adding  and  subtracting  an  undetermined  constant,  the  fractii  m 
may  be  written  in  the  form 

p  cos  x  +  q  sin  x  +  A  (a  cos  x  +  b  sin  x) 
a  cos  x  +  b  sin  x 

we  may  now  assume 
/  cos  x  +  q  sin  x  +  A  (a  cos  x  +  b  sin  x)  =  k  (b  cos  x  —  a  sin  x); 

the  expression  is  then  readily  integrated,  and  A  and  k  so  determin  :d 
as  to  make  the  equation  last  written  an  identity.     The  result  is 

f  p  cos  x  +  q  sin  x   .        ap  +  bq          bp  —  aq .       ,  ,    .      A 

— = —  dx  •=•  -^5 T|  x  +  -^s rf  log  (a  cos  x  +  b  sm  x). 

j  a  cos  x  +  b  sm  x  a  -f  cr  a  +  o 

dx 


*k 


tan  # 


„      7, 
,     See  Ex.  45. 


ax  b       .      ,  ,    r   •      \ 

+    *   ,    *  log  (a  cos  *  4-  £  sin  *). 


,    ,    ,,        *    , 

ff      T"    ™  <r      T 


47.  Find  the  area  of  the  ellipse 

x  —  a  cos  ^  ^  =  b  sin  ^. 


f° 
—  $ab 

' 


sin2  $d$  •=•  nab. 


48.  Find  the  area  of  the  cycloid 

#  =  a       —  sin  F  =  a   i  —  cos 


§  III.]  EXAMPLES.  49 

49.  Find  the  area  of  the  trochoid       (b  <  a) 

x  =  aty  —  b  sin  ^  y  =  a  —  b  cos  ^>. 

(2«2  +  £2)  it. 

50.  Find  the  area  of  the  loop,  and  also  the  area  between  the  curve 
and  the  asymptote,  in  the  case  of  the  strophoid  whose  polar  equation  is 

r  =  a  (sec  0  ±  tan  6). 
Solution  : — 
Using  0  as  an  auxiliary  variable,  we  have 

/          •     \  ,   sin20~| 

x  =  a  (i  ±  sine)  y  —  a\  tan  0  ±  —         , 

L  cos  6J 

the  upper  sign  corresponding  to  the  infinite  branch,  and  the  lower  to 
the  loop.     Hence,  for  the  half  areas  we  obtain 

f*ff  f*ff  f         TT~| 

+  a*\    sin  6  d$  +  a*  \    sin8  6  dQ  =  a*\  i  +  - 

and  —a8  f   sin  9  dB  +  a*  f    sin8  6  <fo  =  a2    i  —  -    . 


5O  METHODS  OF  INTEGRATION.  [Art.  37. 

CHAPTER   II. 

METHODS  OF  INTEGRATION — CONTINUED. 


IV. 

Integration  by  Change  of  Independent  Variable. 

37.  IF  x  is  the  independent  variable  used  in  expressing  an 
integral,  and  y  is  any  function  of  x,  the  integral  may  be  ex- 
pressed in  terms  of  y,  by  substituting  for  x  and  dx  their  values 
in  terms  of  y  and  dy.  By  properly  assuming  the  function  yt 
the  integral  may  frequently  be  made  to  take  a  directly  integra- 
ble  form.  For  example,  the  integral 

f     xdx 


}(ax  +  &)* 
will  obviously  be  simplified  by  assuming 

y  =  ax  +  b 
for  the  new  independent  variable.     This  assumption  gives 

x  —  - ,  whence  dx  =  — ; 

a  a 

substituting,  we  have 

xdx       _£  [Q  —  ft) dy 


§  IV.]  CHANGE   OF  INDEPENDENT   VARIABLE.  5  I 

or  replacing  y  by  x  in  the  result, 

f     x  dx  I  .       ,  b 

~r~    r~A^  =  3  log  \ax  +  o)  +  ~^T—   — TT  • 
i  (ax  +  bj      cr  '      a?  (ax  +  b) 

38.  Again,  if  in  the  integral 

dx 


we  put  y  =  f,    whence 

dv 
x  =  log  y,  and  dx  =  — . 

y 

we  have 


f    dx      _  f       dy 

}e*  —  i  ~~  J          —  i   ' 


y(y 
Hence,  by  formula  (A),  Art.  17, 


It  is  easily  seen  that,  by  this  change  of  independent  variable, 
any  integral  in  which  the  coefficient  of  dx  is  a  rational  func- 
tion of  £*,  may  be  transformed  into  one  in  which  the  coefficient 
of  dy  is  a  rational  function  of  y. 

Transformation  of  Trigonometric  Forms. 

39.  When  in  a  trigonometric  integral  the  coefficient  of  dO  is 
a  rational  function  of  tan  9,  the  integral  will  take  a  rational 
algebraic  form  if  we  put 

tan  6  =  x,  whence  d9  =          »  . 

I  T  X? 


52  METHODS   OF  INTEGRATION.  [Art.  39. 

For  example,  by  this  transformation,  we  have 

f       dB  f  dx 

\  i  J-  tan  0  ~~  J  (i  +^)(i  +x) ' 

Decomposing  the  fraction  in  the  latter  integral,  we  have 

f        dB  I  f    dx       _  i  f  x  dx         i  f   dx 

J  i  +  tan  Q~  ^,]i  +  x*  ~~2\i  +  x*       2J I  +  x 

—  |  tan"  x  —  {  log  (i  +  x*}  +  \  log(l  +  x) 

1  ra      .       I  +  tan  (H 
=  -  \  v  +  log  -  , 

2  L  sec  B    J 

— =  ^  [^  +  log  (cos  0  +  sin  0)]. 


40.  The  method  given  in  the  preceding  article  may  be  em- 
ployed when  the  coefficient  of  dB  is  a  homogeneous  rational  func- 
tion of  sin  B  and  cos  0,  0/  a  degree  indicated  by  an  even  integer ; 
for  such  a  function  is  a  rational  function  of  tan  B.  It  may  also 
be  noticed  that,  when  the  coefficient  of  dB  is  any  rational  func- 
tion of  sin  B  and  cos  6,  the  integral  becomes  rational  and  alge- 
braic if  we  put 

_         0_ 

2* 

for  this  gives 

I+.S2'  ~  I    +  £?'  ~\+£' 

This  transformation  has  in  fact  been  already  employed  in 

the  integration  of -. .     See  Art.  35. 

a  +  b  cos  B 


IV.]      LIMITS   OF   THE    TRANSFORMED    INTEGRAL,  53 


Limits  of  the    Transformed  Integral. 

4-1.  When  a  definite  integral  is  transformed  by  a  change  of 
independent  variable,  it  is  necessary  to  make  a  corresponding 
change  in  the  limits.  If,  for  example,  in  the  integral 

dx 


we  put  x  =  a  tan  6,  whence  dx  =  a  sec2  6  dd, 

we  must  at  the  same  time  replace  the  limits  a  and  co ,  which 
are  values  of  x,  by  \n  and  ^TT,  the  corresponding  values  of  0. 
Thus 

IT 

dx 


f  nrr—v  = 

Ja(a2  +  x*y 


4 

ff 


"   =  -A,  I  0  -f  sin  6  cos  6  \     =     n  Q    . 


Reciprocal  of  x   taken  as  the  New  Independent 
Variable. 

42.  In  the  case  of  fractional  integrals,  it  is  sometimes  use- 
ful to  take  the  reciprocal  of  x  as  the  new  independent  variable. 
For  example,  let  the  given  integral  be 

f       dx 


}**(*  +  i)2' 
Putting  #=-,  whence  dx=. — -3-, 


54  METHODS  OF  INTEGRA  TION.  [Art.  42. 

we  have 

dx  r      y  dy  f    fdy 

^+.y-   j  /  +,\«-  -Jjjny 

\      y) 

Transforming  again  by  putting  z  =  y  +  I,  the  integral  be- 
comes 

(z -  i)3  j  i  (  (dz 


-  3 


Therefore,  since  z  =y  +  i  =  -  +  i  = 


dx 


43.  In  the  above  example  we  see  that  the  single  substitu- 

x  +  I    . 
tion  z  =  -  is  equivalent  to  the  two  substitutions  which 

are  suggested  in  the  process.   In  like  manner,  in  the  case  of  the 
more  general  integral 

f  dx 


(x  —  d}m(x  -  by 

the  successive  transformations  which  suggest  themselves  are 
found  to  be  equivalent  to  a  single  one  in  which  the  new  vari- 
able is 

x  —  a  bz  —  a 

z  =  -  -,,      whence     x  =  -  . 

x  —  b  z  —  i 


§  IV.]  TRANSFORMATION    TO  A    POWER   OF  X.  55 

The  result  is 


which,  when  m  +  n  —  2  is  a  positive  integer,  is  directly  inte 
grable  after  expansion  by  the  Binomial  Theorem. 


A  Power  of  x  taken  as  the  New  Independent  Variable. 
.  The  transformation  of  an  integral  by  the  assumption 


is  not  generally  useful,  since  the  substitution 


L  I    '—i 

x  =  y,  whence  dx  =  -yn    dy, 


will  usually  introduce  radicals.  Exceptional  cases,  however, 
occur.  For,  since  logarithmic  differentiation  of  equation  (i) 
gives 


(2) 


it  is  evident  that,  if  the  expression  to  be  integrated  is  the  product 

dx 
of  —  and  a  function  of  x*,  the  transformed  expression  will  be 

dy 
the  product  of  --  and  the  like  function  of  y. 

For  example,  the  substitution  y  —  x*  transforms 

(x*  -  \)dx    .  (y  -  \]dy 

I)* 


56 


METHODS   OF  INTEGRATION. 


[Art.  44. 


Hence,  decomposing  the  fraction  in  the  latter  expression, 


=:        log 


(*4  +  i)       4 
Again,  putting  ^r3  =  y2  or  jy  =  x*,  we  have 


_2f 

-<?  =  ~ 


_  _t_ 

=      s    """8    "* 


4-5.  When  this  method  is  applied  to  an  integral  whose  form 
at  the  same  time  suggests  the  employment  of  the  reciprocal, 
as  in  Art.  42,  we  may  at  once  assume  y  =  x~*.  Thus,  given 
the  integral 


x  X*(2  +  X3)  ' 


putting 
we  obtain 


y  =  x~*>  whence 


dx  _        dy 
~ 


ydy 


I,  2y  4-  i 

=      y      log(2j+  I)"]0=2-log3 
6  12  Jx  12 

Examples  IV. 


§  IV.]  EXAMPLES.  57 

2X  —    I 

!> 


far1  —  ^r  +  i 
'   J  (2*  +  i)2        ' 


dx 


f 
J 


2.r  +  i        log  (2.*  +  i)  _          7 
~~  ~~          ~ 


-  log 


1  .     e*  —  i 
-  log  — 

2  8^  +   I 


2  log  ^  ~  l) 

/•^ 

f  2  +  tan  6  6  —  log  (3  cos  6  —  sin  9) 

o.    I  •  —  ~  -  u9«  ~  .       .    .         _,_ 

J  3  —  tan  9  2 

tan  9  —  i       9 


f       dQ  J_ 

I0>  Jtan29-i'  4  °g 


tan  9  +  i      2 


f  tan"  9  dQ  £  .      tan  9  —  i       9 

'  Jtan29  -i'  4     gtan9+i       2 


cos  9  ^9  aft  —  b  log  (g  cos  9  —  ^  sin  e) 

a  cos  9  —  £  sin  9  '  «a  +  b* 


$8  METHODS  OF  INTEGRATION.  [Ex.  IV. 


f     COS  QdQ 

-  r  -  T7T' 

J  COS  (a  +  6) 

(6  +  a)  cos  a  —  sin  a  log  cos  (9  +  a). 


sin  (0  +  a) 


(6  +  /3)  cos  (a  —  /3)  +  sin  (a  —  /3)  log  sin  (0  +  /?). 
15.      tan  (0  +  ct)  cos  0  */0,    —cos  0  +  sin  a  log  tan 


29   +   2fX  +  7t 


,     fa     cos  Q  dQ  .  . 

16.       -i  —  -  --  —  -  r,  cos  «  log  (2  cos  a)  +  asm  a. 

J0  sin  (a  +  6) 

—  * 

p"  cosje  ^fl  _i_         4/2+2  sin  flH  6  _  log  (3  +  2  ^2) 

Jo    cose    '  4/2     °  -/2  —  2  sinfi'Jo  4/2 

fsinJ^Q  ^  -f  Q 

18.       —  ^—  —  ,  log  tan  --  . 

J     sin  Q  4 


XT  aX 


f°°    ^*<£r  i  (7  .  ,  ^ 

21.       7  —      ,.8>  —      sm  20  //9  =  -7- 

Jo  (i  +  ^r)  4Jo  16 


§  IV.]  EXAMPLES.  59 


f       dx 

1    +  '        los  *  +  ' 

23-    L.S  /„  ,   T\i         * 

J-*  \^  <   -1  ) 

f        <£r 

n  A         1  

I                       I                         or 

.  J  -1-  Inrr  

I   —  X 


2  -  log  3 


</AT  I  X 

'•  4'  g 


V. 

Integrals  Containing  Radicals. 

46.  An  integral  containing  a  single  radical,  in  which  the 
expression  under  the  radical  sign  is  of  the  first  degree,  is 
rationalized,  that  is,  transformed  into  a  rational  integral,  by- 
taking  the  radical  as  the  value  of  the  new  independent  vari- 
able. Thus,  given  the  integral 

f          dx 

J  i  +  V(x+  i)' 


60  METHODS  OF  INTEGRATION.  [Art.  46. 


putting                                y  =  4/(. 

X  +    I), 

whence            x=y*—\, 

and 

dx  —  2y  dyt 

we  have 

f       dx       ^~\y*y  ._  ,( 

ffv  —  2  1  ¥— 

i)        Ji  +  y        J  '         Ji  +y 

=  2y-  2  log  (i  +y) 

=  2V(x  +  i)-  2log  [i  +  V(x  +  i)]. 

47. 'The  same  method  evidently  applies  whenever  all  the 
radicals  which  occur  in  the  integral  are  powers  of  a  single 
radical,  in  which  the  expression  under  the  radical  sign  is  linear. 
Thus,  in  the  integral 

dx 


the  radicals  are  powers  of  (x  —  i)£  ;  hence  we  put  y  =  (x  — 
and  obtain 

dx 


48.  An  integral  in  which  a  binomial  expression  occurs 
under  the  radical  sign  can  sometimes  be  reduced  to  the  form 
considered  above  by  the  method  of  Art.  44.  For  example, 
since 

f        dx 


§  V.]  INTEGRALS  CONTAINING  RADICALS.  6 1 

fulfils  the  condition  given  in  Art.  44  when  n  =  3,  the  quantity 
under  the  radical  sign  may  be  reduced  to  the  first  degree. 
Hence,  in  accordance  with  Art.  46,  we  may  take  the  radical  as 
the  value  of  the  new  independent  variable.  Thus,  putting 


o 
whence  a?  •=.  ?  —  i,  and 

-Afc'-O* 


i)4, 

dx         4^3  dz 


—       -        —  -., 

-  * 

we  have 


dx        _  4  (£  dz 


Decomposing  the  fraction  in  the  latter  integral  (see  Art.  22), 
we  have  finally 


-  =  i  tan-<[V  +  i)*~|  -f  1 
I)i      3  'J      3 


Radicals  of  the  Form  -<j(ax*  +  3). 

49.  It  is  evident  that  the  method  given  in  the  preceding 
article  is  applicable  to  all  integrals  of  the  general  form 


(i) 


in  which  m  and  n  are  positive  or  negative  integers.     These 
integrals  are  therefore  rationalized  by  putting 


y  = 


62  METHODS  OF  INTEGRATION.  [Art.  49. 


Putting  m  =  O,  the  form  (i)  includes  the  directly  integrable 
case 

+  b**  xdx. 


50.  As  an  illustration  let  us  take  the  integral 
f          dx 


putting  y  =  Vx   +  a), 

whence  x*=yi  —  a2>  and  •  —  =  •  0        *, 

x      f  —  a* 

we  have 

dx          __  f     dy 


Hence,  by  equation  (A')  Art.  17, 

f          dx  _   i   .       y  —  a        I          ^(x2  +  cF)  —  a 

J  x V(x*  +  #2)      2a        y  +  a      2a        V(x*  +  a?)  +  a' 


Rationalizing  the  denominator  of  the  fraction  in  this  result, 
we  have 

V(x*  +  a2)  -  a  _  [  V(x*  +  a*)  -  aj 
V(x*  +  a*)  +  a  ~  x* 

Therefore 


V.]  INTEGRALS   CONTAINING  RADICALS.  63 

In  a  similar  manner  we  may  prove  that 
dx  i         a  — 


51.  Integrals  of  the  form 

......     (2) 


are  reducible  to  the  form  (i)  Art.  49,  by  first  putting  y  =  -. 

X 

For  example ; 

dx 


is  of  the  form  (2)  ;  but,  putting  x  —  -  ,  whence 


+  6)=  rv"  ^VJ>  >  and  dx  =  -^ 

y  j 

we  obtain 

f        dx  f      y  dy 


(ax2  +  b)*  J  (a  + 


The  resulting  expression  is  in  this  case  directly  integrable. 
Thus 

f       dx  i  x  .  r, 

.     .     .     \j  ) 


*      b  V  (a  +  bf)      b\f(ax*  +  b) 


64  METHODS  OF  INTEGRATION.  [Art.  52. 

Integration  of  — 7— ^ ^- . 

A /I  -y*  +  //    i 
^/  i  .*<     ^±-  u    i 

52.  If  we  assume  a  new  variable  z  connected  with  x  by  the 
relation 

we  have,  by  squaring, 

£  —  2.zx  =  ±  cP, (2) 

and,  by  differentiating  this  equation, 

2  (z  —  x)  dz  —  2z  dx  =  o ; 
whence 

dx        dz 


Z  —  X         2 

or  by  equation  (i), 


dx  dz  t  ^ 

r»     ......    (3) 


±0*)      2' 
Integrating  equation  (3),  we  obtain 

>     •    •    (AT) 


53.  Since  the  value  of  x  in  terms  of  z,  derived  from  equa- 
tion (2)  of  the  preceding  article,  is  rational,  it  is  obvious  that 
this  transformation  may  be  employed  to  rationalize  any  ex- 
pression which  consists  of  the  product  of  - •  .  , — -,.  and  a 
rational  function  of  x.  For  example,  let  us  find  the  value  of 


§  V.]  TRIGONOMETRIC   TRANSFORMATION.  65 

which  may  be  written  in  the  form 


By  equation  (2) 


whence 


Therefore,  by  equations  (3)  and  (5), 

f    ..  ,  i  f( 

VC^3  ±  or)  dx  =  - 


i        ,     , 
± 
4J  2 


(dz      a*  (dz 

-  +  - 
J  z       4  )  zr 


a* 
±  - 


By  equations  (4)  and  (5),  the  first  term  of  the  last  member 
is  equal  to  \  x  V(x*  ±  «2)-     Hence 


^)].    .    (L) 


Transformation  to   Trigonometric  Forms. 

64.  Integrals  involving  either  of  the  radicals 

x*  or 


66  METHODS  OF  INTEGRATION.  [Art.  $4. 

can  be  transformed  into  rational  trigonometric  integrals.     The 
transformation  is  effected  in  the  first  case  by  putting 

x  —  a  sin  6,  whence  V(a*  —  X2)  =  a  cos  6  ; 

in  the  second  case,  by  putting 

x  =  a  tan  8,  whence  V(az  +  x*)  =  a  sec  6  ; 

and  in  the  third  case,  by  putting 

x  =  a  sec  0,  whence  ^(x*  —  a?)  =  a  tan  6. 

55.  As  an  illustration,  let  us  take  the  integral 


putting  x  =  a  sin  0,  we'have  V(#,  —  x2)  =  a  cos  0,dx  =  a  cos  B  d0- 
hence 


f  V(<#  -x*}dx  =  a>  [ 


COSZ0d0 

___  a2  0      c?  sin  0  cos  0 
by  formula  (C)  Art.  29.     Replacing  0  by  x  in  the  result, 


//-2  ~2\    -7  •  —  ,  ,,x 

V  (a*  —  ^r)  dx  =  —  sin  - x  -  H ~ .     .     .     (M) 

J  2  d  2 

Regarding  the   radical    as   a  positive   quantity,  the  value 
of   0  may  be   restricted  to  the  primary  value  of  the  symbol 

% 
sin-1-  (see  Diff.  Calc.,  Art.  57);  that  is,  as  x  passes  from  —  a 

to  +  a,  0  passes  from  —  |  TT  to  +  £  n. 


§V.]  INTEGRALS  CONTAINING  RADICALS.  6? 


Radicals  of  the  Form  ^/(a^  +  bx  +  c). 


56.  When  a  radical  of  the  form  y(a^  +  bx  +  c)  occurs  in  an 
integral,  a  simple  change  of  independent  variable  will  cause  the 
radical  to  assume  one  of  the  forms  considered  in  the  preceding 
articles.  Thus,  if  the  coefficient  of  X*  is  positive, 


in  which,  if  we  put.r+  —  =  v,   the   radical   takes   the    form 

2a 


+  tf2)  or  \f(yi  —  tf2),  according  as  ^ac  —  &  is  positive  or 
negative.  If  a  is  negative,  the  radical  can  in  like  manner  be 
reduced  to  the  form  y(a?  —  y*)  orV(  —  cP—y*}  ;  but  the  latter  will 
never  occur,  since  it  is  imaginary  for  all  values  of  y,  and  there- 
fore imaginary  for  all  values  of  x. 

For  example,  by  this  transformation,  the  integral 

p  dx 


J  (a*  +  bx  +  <:)! 

can  be  reduced  at  once  to  the  form  (J\  Art.  51.    Thus 

dx 


+ 




(4ac '  ^(a* +  bx + c)' 


68  METHODS  OF  INTEGRATION.  [Alt.  5/. 

57.  When  the  form  of  the  integral  suggests  a  further 
change  of  independent  variable,  we  may  at  once  assume  the 
expression  for  the  new  variable  in  the  required  form.  For 
example,  given  the  integral 


V(2ax  —  x*}  x  dx ; 


we  have  \f(2ax  —  x*)  =  V[a?  —  (x  —  of] 

hence  (see  Art.  54),  if  we  put  x  —  a  =  a  sin  0,  we  have 

V(2ax  —  x2)  =  a  cos  6, 
x  ==  a  (i  +  sin  0),  dx  =  a  cos  &  dO ; 

.•\V(2ax  -  x*}xdx  =  c?  fcos2  0(i  +  sin  0)  dO 

/T3  /7^ 

C*  /  yj  •  /)  /1\  €^»-  q    /J 

=  —  (6  +  sin  0  cos  0) coss  0 

2    V  3 

=  —  sin~J -\ — (x  —  a)  V(2ax  —  x*) (2ax  — 

2  a          2V  3  v 

Ct         ,  X  ~~"  d          1        .f  o\  r        9 

=  —  sin-1 h  7-  V(2ax  —  x*)  yzx2-  —  ax  — 

2  a         o 

The  Integrals 
dx  ,     r  dx 


58.  An  integral  of  the  form     —-, — ^ — j r-  may  by  the 

J  V(eur  +  bx  +  c] 

method  of  Art.  56,  be  reduced  to  the  form  (K),  Art.  52,  or  to 
the  form  (/'),  Art.  10,  according  as  a  is  positive  or  negative. 


§  V.]  IRRATIONAL   INTEGRALS.  69 

But,  when  the  integral  appears  in  one  of  the  above  forms  (the 
quadratic  under  the  radical  sign  admitting  of  linear  factors), 
another  mode  of  transforming  is  often  convenient. 

Assuming  ft  >  a,  we  may  in  the  first  case,  since  the  differ- 
ence of  the  factors  is  the  constant  ft  —  a,  put 

x  —  a.  —  (ft  —  a)  sec2  6  \ 
x  -  ft  =  (ft  -  a)  tan2  0  j  ' 

whence 

dx  =  2(fi  —  a)  sec2  0  tan  0dBt 
and 

j        -  *L  -  _  =  2\sec0ii0  =  2  log  (sec  0  4-  tan  0). 
J  </[(*  -  «)(*  -  /*)]         J 

Substituting,  and  omitting  the  constant  log  \f(ft  —  a), 


In  the  second  case,  the  sum  of  the  factors  being  ft  —  a,  we 
may  put 


—  of  =  (ft  —  a)  sin2  0  ) 
-  x  =  (ft  -  a)  cos2  0  )' 


x  —  of  =       — 
ft 

whence 

dx  =  2(/3  —  a)  sin  0  cos  0d0, 

and  the  quantity  under  the  integral  sign  reduces  to  2dO.    There- 
fore 


The  same  transformations  may  of  course  be  employed  when 
other  factors  occur  in  connection  with  these  radicals 


7O  METHODS  OF  INTEGRATION.  [Art.  58. 

It  can  be  shown  that  the  values  given  in  formulas  (TV)  and 
(O)  differ  only  by  constants  from  the  results  derived  by  em- 
ploying the  process  given  in  Art.  56. 


Examples  V. 

.      V(a  —  x)-x  dx,  --  (a  —  x)%  ($x  +  20). 

J  0 


2 


f    xdx  2     3 

3-      T  >  -  *    —  x  + 

J  i  +  Vx'  3 


.     [     /^        ,  2  V*  +  2  log  (l  - 

J   T*  —   I 


r  dx  2  /2JC  ~—  # 

7.       -r-      -TV,  -tan"1  j/- 

jxV(2ax  —  a)  a  a 


, 
8.        (a— 


f       ^ 

'liA 

J  2^  —  x* 


9  7  5    JvA*       315 

H 1 1 Q • 

44  8 


§  V.]  EXAMPLES. 


12.    I  -  -7—  ^ 
}x  —  M(x*  — 


f  V(x<  +  i 
J5-  J        -£ 


.6.   f(*-+'"' 

X 


3-f  -  ^T  =  —  +  A- 

8         5  Ji  10        40 


Rationalize  the  denominator. 


2  (x  +  a)*  -  2  (x  + 
3  («  -  *) 


I         j/(jg*  +  i)  —  i 

4  og  i/(^4  +  i)  + 


tan- 


v 

—  a8)  —  a  sec-1  - 


72  METHODS  OF  INTEGRATION.  [Ex.  V. 


—        ,x   3     —  ^-dx.     See  formulas,  (L)  and  (K\ 


-xV(x*  +  «*)--  0s  log  [x 


20.  — -  dx,  a  log  — 


dx 

21. 


/%  n  _  /V-y* 

J*i.         I  ^^   UvV* 

»  t-V 


dx 
23 


i        r    //    a     ,       2\  n          ^(^  +  a*) 

log  [  4/(^a  +  a2)  +  j;]  --  ^—  -  -  ' 


*  J  VCr'  +  «f )  -  a  ' 

4/(^ra  +  a*)        « 


i 
log 


X  X 


f         •£  /TY  T 

24.  J  y(l  +  g«)  •     ^  -«w»»/tf  (^).     jlog  [^8  +  l/(i  -f  a;4 


§  V.]  EXAMPLES.  73 

25.   I  V(ax*  +  b]  dxt          [a  >o]         Put  V(ax*  +  b)  =  z  —x^a. 


b  i 

— -  log  [x  4/0  +  tf(ax   +  b}\  +  - 

V d  2 


26.   [,- 

}(a 


dx 


+  x)  ^(x*  +  #}  ' 


28.  -^ 


29. 


dx 


dx 


i     x 
g 


d&c  V(-^a  ~  J)  i  L 

7    »  7*  Q  "i 


2JC  2 


log  tan  ^ 

8         2 


74  METHODS  OF  INTEGRATION.  [Ex.  V. 


f          dx  /     2         \ 

•k-vt**-.)'  ("+l) 


oo* 


+ 
34- 


__         j 

ta 


,, 

3 


37' 


38- 


39 


(a      //  s\    ^  <?7t 

.        ^(tor  —  «*)H^  -— 

jo  4 

f* 

40.  I     V(2aA:  —  x*)'X  dxt 

Jo 

0s  |    w  cos80  (i  +  sin  6)  &  =  a'\  -  —  -  1 

2 

41.  I     tf(2ax  —  x*)'X*  dxt 

Jo 

a4  f°  w  cos'6  (i  +  sine)8  <to  =  a*  f"^  -  -1 


§  V.]  EXAMPLES.  75 


dx 


by  Art.  56,         log  [x  +  a  +  V(zax  +  x*)]  +  C; 
by  Art.  58,  2  log  [  V*  +  V(2a  +  x)]  +  C. 


•  T     •*     ^"~    ^*  ,/ 

sin-    --  V(2ax  — 


4,     f_^  -  ^  -  _  £y  ^r/.  56,  sin-  r  X-^-^  +  C 

45    J  V($  +  4^  -  ^)  3 


by  Art.  58,            2  sin-1          j     +  C' 

t*          dx  ,/x~\a 

,6        _  —  -  .  asin-W-        =?r 

'•  J0  V(oy-^)»  K  «J0 


49. 
y 


•     ,x-i      (x  +  3)  V(3 

>  3  sm"  - 


—,, 
—ax)' 


?6  METHODS  OF  INTEGRATION.  [Ex.  V. 

50.  Find  the  area  included  by  the  rectangular  hyperbola 

y  =  2ax  +  x3, 
and  the  double  ordinate  of  the  point  for  which  x  =  2a. 

a*[6  V2  —  log  (3  +  2  ^2)]. 

51.  Find  the  area  included  between  the  cissoid 

x  (x9  +  y )  =  2ay* 

and  the  coordinates  of  the  point  (a,  a)  ;  also  the  whole  area  between 
the  curve  and  its  asymptote. 


.         a--)-. 

52.  Find  the  area  of  the  loop  of  the  strophoid 


and 


=  o 


also  the  area  between  the  curve  and  its  asymptote. 


za  (  i )  ,       and        20* 

4 


For  the  loop  put  y  =  —  x     .  3 -j-  ,  since  x  is  negative  between  the  limits 

\f  \Cl     ^~  ^v   J 

—  a  and  o. 

53.  Show  that  the  area  of  the  segment  of  an  ellipse  between  the 

x 

minor  axis  and  any  double  ordinate  is  ab  sin  - l  — I-  xy. 

a 


§  VI.  (  INTEGRATION  BY  PARTS.  77 

VI. 

Integration  by  Parts. 

59.  Let  u  and  v  be  any  two  functions  of  x ;  then  since 
d  (uv)  =  u  dv  +  v  du, 

uv  —  \u  dv  +  \v  du, 
\u  dv  =  uv  —  \v  du ( i ) 


whence 


By  means  of  this  formula,  the  integration  of  an  expression 
of  the  form  udv,  in  which  dv  is  the  differential  of  a  known 
function  v,  may  be  made  to  depend  upon  the  integration  of 
the  expression  v  du.  For  example,  if 

^^cos"1-*"          and         dv  =  dx, 
we  have 

,  dx 


hence,  by  equation  (i), 

f  j  ,  f      *dx 

cos'1  x-dx  =  ^rcos-1^-  +    - 

]  }  V(i  -  x*y 

in  which  the  new  integral  is  directly  integrable  ;  therefore 

cos-1^-^  =  ^rcos'1^  —  V(i  —  ^2). 
The  employment  of  this  formula  is  called  integration  by  parts. 


METHODS  OF  INTEGRATION. 


[Art.  60. 


Geometrical  Illustration. 

60.  The  formula  for  integration  by  parts  may  be  geomet- 
rically illustrated  as  follows.     Assum- 
ing rectangular  axes,  let  the  curve  be 
constructed  in  which  the  abscissa  and 
ordinate  of  each  point  are  correspond- 
ing values    of   v  and  u,  and   let    this 
curve  cut  one  of  the  axes  in  B.     From 
any  point   P  of  this   curve  draw   PR 
r(    and   PS,   perpendicular    to  the   axes. 
1   Now  the  area  PBOR  is  a  value  of  the 

indefinite  integral  \u  dv,   and    in    like 
manner  the  area  PBS  is  a  value  of  \vdu ; 
and  we  have 


B 

A 

o 


Area  PBOR  =  Rectangle  PSOR  -  Area-fiRS; 


therefore 


\u  dv  =  uv  —  \v  du. 


Applications. 

61.  In  general  there  will  be  more  than  one  possible  method 
of  selecting  the  factors  u  and  dv.  The  latter  of  course  in- 
cludes the  factor  dx,  but  it  will  generally  be  advisable  to  in- 
clude in  it  any  other  factors  which  permit  the  direct  integra- 
tion of  dv.  After  selecting  the  factors,  it  will  be  found  con- 
venient at  once  to  write  the  product  u-v,  separating  the  factors 
by  a  period ;  this  will  serve  as  a  guide  in  forming  the  product 


§  VI.]  INTEGRATION  BY  PARTS.  79 

v  du,  which  is  to  be  written  under  the  integral  sign.     Thus,  let 
the  given  integral  be 


J*Mog 


x  dx. 


Taking  x*  dx  as  the  value  of  dv,  since  we  can  integrate  this 
expression  directly,  we  have 

\X*\Q&X  dx  =  log  x-  —  x* \x^  — 

J  3  3-1       x 

=  —  x*  log  x \x*dx 

3  3J 

x* 
=  -(3  log*-  i). 

(V 

62.  The  form  of  the  new  integral  may  be  such  that  a 
second  application  of  the  formula  is  required  before  a  directly 
integrable  form  is  produced.  For  example,  let  the  given 
integral  be 

x*  cos  x  dx. 

In  this  case  we  take  cos  x  dx  =  dv ;  so  that  having  x*  =  u,  the 
new  integral  will  contain  a  lower  power  of  x :  thus 

\x*  cos  xdx  =  ^-sin  x  —  2  LF sin  xdx. 
Making  a  second  application  of  the  formula,  we  have 
\x*  cos  x  dx  =  x*  sin  x  —  2\  x(-  cos  x*)  +    cos  xdx  I 
=  ^sin  x  +  2x  cos  x  —  2  sin  x. 


8O  METHODS  OF  INTEGRATION.  [Art.  63. 

63.  The  method  of  integration  by  parts  is  sometimes 
employed  with  advantage,  even  when  the  new  integral  is  no 
simpler  than  the  given  one  ;  for,  in  the  process  of  successive 
applications  of  the  formula,  the  original  integral  may  be  repro. 
duced,  as  in'  the  following  example: 

emx  sin  (nx  +  a)  dx 

—  cos  (nx  +  a)      m  t  . 

=  ewx  '  -  —  -  -\  ---  emx  cos  (nx  +  a)  dx 

n  } 

{ 

\ 
J 


n 


emx  cos  (nx  +  a)     m        sin  (nx  +  a)      mz  { 
=  --  —*-  +  -em*-  —+  --     e"!X 

n  n  n 


in  which  the  integral  in  the  second  member  is  identical  with 
the  given  integral  ;  hence,  transposing  and  dividing, 

[  emx 

\  vnx  sin  (nx  +  a)  dx  =     2        «  [m  sin  (nx  +  a)  —  n  cos  (nx  +  a)]. 


64.  In  some  cases  it  is  necessary  to  employ  some  other 
mode  of  transformation,  in  connection  with  the  method  of 
parts.  For  example,  given  the  integral 


taking  dv  =  sec2  0  dd,  we  have 

[sec8  0  <#  =  sec  0-tan  0  -  fsec0tan20^#.     .     .     (i) 


§  VI.]  FORMULAE  OF  REDUCTION.  8 1 

If  now  we  apply  the  method  of  parts  to  the  new  integral,  by 
putting 

sec  6  tan  6  dd  =  dv, 

the  original  integral  will  indeed  be  reproduced  in  the  second 
member ;  but  it  will  disappear  from  the  equation,  the  result 
being  an  identity.  If,  however,  in  equation  (i),  we  transform 
the  final  integral  by  means  of  the  equation  tan2  9  =  sec2  0  —  \, 
we  have 

[sec3  Od6=  sec  6  tan  B  -  [sec8  8  dB  +  f  sec  BdB. 
Transposing, 

f  «    n      in  Sin   B  [      dd       j/i 

2   sec8  0  dO  =  — j-^  + -dO: 

J  cos2  0      J  cos  a 

hence,  by  formula  (F),  Art.  31, 

f      ,/,,/,         sin  B        i  ,  \~n      6~\ 

sec8  BdB  —  5-2  +  -  log  tan     -  +  -    . 

J  2cos2#      2  l_4      2j 

Formula  of  Reduction. 

65.  It  frequently  happens  that  the  new  integral  introduced 
by  applying  the  method  of  parts  differs  from  the  given  integral 
only  in  the  values  of  certain  constants.  If  these  constants  are 
expressed  algebraically,  the  formula  expressing  the  first  trans- 
formation is  adapted  to  the  successive  transformations  of  the 
new  integrals  introduced,  and  is  called  a  formula  of  reduction. 


82  METHODS  OF  INTEGRATION.  [Art.  6$. 

For  example,  applying  the  method  of  parts  to  the  integral 

[xne**dx, 
we  have 

t  £O.X  ft      t 

\xneaxdx  —  xn \xn-*eaxdx.    .     .     .     .     (l) 

J  a       a] 

in  which  the  new  integral  is  of  the  same  form  as  the  given 
one,  the  exponent  of  x  being  decreased  by  unity.  Equation 
(i)  is  therefore  a  formula  of  reduction  for  this  function.  Sup- 
posing n  to  be  a  positive  integer,  we  shall  finally  arrive  at  the 

f  eax 

integral  eajr  dx,  whose  value  is — .  Thus,  by  successive  appli- 
cation of  equation  (i)  we  have 


Reduction  of  \sinmBdQand  \cosmQdQ. 

66.  To  obtain  a  formula  of  reduction,  it  is  sometimes  neces- 
sary to  make  a  further  transformation  of  the  equation  obtained 
by  the  method  of  parts.  Thus,  for  the  integral 


the  method  of  parts  gives 


§  VI.]      REDUCTION  OF    TRIGONOMETRIC  INTEGRALS.  83 

Substituting  in  the  latter  integral  I  —  sin2  6  for  cos2  d, 
[sin™  Odd  =  —  sin™-1  8  cos  8 

+  (m  -  i)  (smm~*  6  d6  —  (m  -  i)  fsinw  0  dd  ; 

transposing  and  dividing,  we  have 

f    .       n  ,Q          sin"2-1  6  cos  6      m  -  i  f  .  . 

sinw0^0=i  ---  1  --  smw-20^6>,    .     .     .     (i) 
J  mm} 

a  formula  of  reduction  in  which  the  exponent  of  sin  6  is  dimin- 
ished two  units.  By  successive  application  of  this  formula,  we 
have,  for  example  : 


f  •  B  /i  JQ          sm*  #  cos  #      5  f   • 
sin6  0^0=  --  g  --  +  g-l  sin4 


64 


sin5  0  cos  0  _  5  sin3  6  cos  0      5-3  sin  0  cos  0      5-3-1 
6  6-4  6-4-2  6-4-2 

67.  By  a  process  similar  to  that  employed  in  deriving 
equation  (i),  or  simply  by  putting  6  =  £TT  —  6'  in  that  equa- 
tion, we  find 


,  (2) 


mm 


a  formula  of  reduction,  when  w  is  positive. 


84  METHODS  OF  INTEGRATION.  [Art.  68. 

68.  It  should  be  noticed  that,  when  m  is  negative,  equation 
(i)  Art.  66  is  not  a  formula  of  reduction,  because  the  exponent 
in  the  new  integral  is  in  that  case  numerically  greater  than  the 
exponent  in  the  given  integral.      But,  if  we  now  regard  the 
integral  in  the  second  member  as  the  given' one,  the  equation 
is  readily  converted  into  a  formula  of  reduction.     Thus,  put- 
ting —  n  for  the  negative  exponent  m  —  2,  whence 

m  —  —  n  +  2, 
transposing  and  dividing,  equation  (i)  becomes 

f  dO         _          cos  0  n  —  2  f     d0  .  . 

}  sin"  6  ~       (n—  ^sin"-1  0  +  n  —  i  Jsin*-20' 

Again,  putting  6  =  ^  n  —  6'  in  this  equation,  we  obtain 

f   d6     _  sin  8  n  —  2  f     dO 

}  cos*  6  ~  (n  —  i)  cos*-1^  +  n—  i  Jcos*-20 

Reduction  of  \sinmB  cos"  9  d0. 

69.  If  we  put  dv  =  sin"*  6  cos  6  d0,  we  have 

cos"-1/?  sin"' +'0 


in™  0cos*  9d9  = 


m  +  i 


72  j     f 

*  m  +  i  J  S 

but,   if  in  the  same  integral  we  put  dv  =  cos"  0  sin  0d0t  we 

have 

'0 


«  +  i 
—  i 


.    ...    (2) 


§  VI.]       REDUCTION  OF    TRIGONOMETRIC  INTEGRALS.  85 

When  m  and  n  are  both  positive,  equation  (i)  is  not  a 
formula  of  reduction,  since  in  the  new  integral  the  exponent 
of  sin  6  is  increased,  while  that  of  cos  B  is  diminished.  We 
therefore  substitute  in  this  integral 

sinw+2  6  =  sinw  6  (i  —  cos2  #), 
so  that  the  last  term  of  the  equation  becomes 

n~  l  fsin'"  B  cos"-2  BdB-  n~  l    \  sin"1  8  cos«  BdB. 
m  +  i  J  m  +  i  J 

Hence,  by  this  transformation,  the  original  integral  is  repro- 
duced, and  equation  (i)  becomes 


[~l  + -1 1  f  sin'"  0  cosn8d8  = 

m  +  IJJ 


m  +  I 


«-  I  f  .  , 

sin* 
m  +  i  j 


T>..  ...       ,  n  —  I        m  +  n  , 

Dividing  by  i  ^ = ,  we  have 

&    *          m  +  i       w  +  i 


f  • 
si 

J 


•  ~a       *aja 
sin**  6  cos"  6d8= 


m  + 


rc~  T    [sm^^cos"-2(9^,    ...     (3) 
«  +  n  J 


a  formula  of   reduction   by  which  the  exponent  of  cos  B  is 
diminished  two  units. 


B6  METHODS  OF  INTEGRATION.  [Art.  69. 

By  a  similar  process,  from  equation  (2),  or  simply  by  put- 
ting 0  =  %  n  —  6'  in  equation  (3),  and  interchanging  m  and  n, 
we  obtain 

f    •    ~n         «nJa  sin**-1  d  COSn+1  B 

sin**  6  cos"  B  dd  = 
J 


m  +  n 


m- 
T#  -I- 


i   fsinw_2  ^  CQS«  ede  ,  , 

«  J 


a  formula  by  which  the  exponent  of  sin  6  is  diminished  two 
units. 

70.  When  n  is  positive  and  m  negative,  equation  (i)  of 
the  preceding  article  is  itself  a  formula  of  reduction,  for  both 
exponents  are  in  that  case  numerically  diminished.  Putting 
—  m  in  place  of  m,  the  equation  becomes 

fcos*#  ,._  __  cos"-1^  n—  i   fcos*-2# 

J  sin™  6  ~  (m—  ^sin^-'tf      m—  i  }smm-2(T  '     '     '     '$' 

Similarly,  when  m  is  positive  and  n  negative,  equation  (2)  gives 

f  sin"*  0   ,Q  _        sin^-'g  m—  i   fsnV"-20 

~  (w  —  i)cos"-1  6      n  ^-  i  Jcos"-2^ 


cos 


71.  When  m  and  n  are  both  negative,  putting  —  m  and  —  n 
In  place  of  m  and  »,  equation  (3)  Art.  69  becomes 


f_ 

J  sinw 


dO 


__  _ 
B  cos"  B  ~~       (m  +  n)  sinm-IB  cos"+1  B 


+ 


n  +  i  f 

w  4-  «  J  sii 


s'mm0cosn+*0t 
in  which  the  exponent  of  cos  0  is  numerically  increased.     We 


§  VL]      REDUCTION  OF   TRIGONOMETRIC  INTEGRALS,  87 

may  therefore  regard  the  integral  in  the  second  member  as  the 
integral  to  be  reduced.  Thus,  putting  n  in  place  of  n  +  2,  we 
derive 


J  sin™ 


d6 


0cos*0      (n  —  i)  sin**-1  0  cos*-1  0 

m  +  n  —  2  f          dO  ,  . 

»-  i        sin-"-2  V7' 


Putting  0  =  £  TT  —  0',  and  interchanging  m  and  »,  we  have 
dO  i 


.  . 

"     ' 


__ 
«  0~      (m—i)s\n.m 

m  +  n  —  2  f  '        dO 


m—i 


72.  The  application  of  the  formulae  derived  in  the  preced- 
ing articles  to  definite  integrals  will  be  given  in  the  next  sec- 
tion. When  the  value  of  the  indefinite  integral  is  required,  it 
should  first  be  ascertained  whether  the  given  integral  belongs 
to  one  of  the  directly  integrable  cases  mentioned  in  Arts.  27 
and  28.  If  it  does  not,  the  formulae  of  reduction  must  be 
used,  and  if  m  and  n  are  integers,  we  shall  finally  arrive  at  a 
directly  integrable  form. 

As  an  illustration,  let  us  take  the  integral 

[  sin2  0  cos4  6  dd. 

Employing  formula  (4)  Art.  69,  by  which  the  exponent  of  sin  6 
is  diminished,  we  have 


8  cos'  8M  =  -  +       cos-  »  dB. 


88  METHODS  OF  INTEGRATION.  [Art.  72. 

The  last  integral  can  be  reduced  by  means  of  formula  (2)  Art. 
67,  which,  when  m  =  4,  gives 

f       4/17/1       cos3  6  sin  6      3  f      „  _   _„ 
cos4  6  dO  =  —  -  +  -    cos2  8  dO  ; 

J  4  4  J 

therefore 

f  •   o  a       A  a  jo          sin#cos50      cos3#sin#  ,  sin  6  cos  0       6 
sin2  0  cos4  Odd  =  --  -  --  1  ---  1  --  -  --  \-  —., 
}  6  24  16  16 

73.  Again,  let  the  given  integral  be 


cos 


J     sin30 

By  equation  (5),  Art.  70,  we  have 

f  cos6  d  dS  _        cos5  6       5  fcos4  8  dtt 
J     sin3  6  2  sin2  6      2  J     sin  6 

We  cannot  apply  the  same  formula  to  the  new  integral,  since 
the  denominator  m  —  i  vanishes  ;  but  putting  n  —  4  and  m  —  —  i, 
in  equation  (3)  Art.  69,  we  have 


fcos4  OdS  _  cos3fl       fcos 
J    sin  6  J    si 


3  sin  Q 


cos3>?       f   dO          {    . 
-     -  +    -  —  5-  —     sin  Odd 
3          Jsm  (?       J 


A         ,-. 

cos3  0      ,  I  ^  ,, 

+  log  tan  —  6  +  cos  0. 


3  2 

Hence 


fcos6  6dd  cos5  0       5  cos8  6       5.  i          5 

-T-S-7T-  ==  --  ^^Q  —   —  z  --  -  l°g  tan  -  0  —  ~  cos  *• 
J     sin8  0  2  sin2  0  6  2     B        2         2 


§  VI.]  REDUCTION   OF   TRIGONOMETRIC  INTEGRALS.  89 

Reduction  of  I  cosm&  cos  nO  dQ. 
74-.  Integrating  by  parts,  we  have 

cos"*  Ocosn6d&=cosm  6  .  --  h  —    sin  »0cosm~I0sin  6dB. 
J  n          n  ] 

To  obtain  a  formula  of  reduction  we  employ  the  identity 

sin  nB  sin  d  =  cos  (n  —  i)#  —  cos  ##  cos  6, 
so  that  the  last  term  of  the  equation  becomes 

Wl  f  W'l  f 

cos'—'fl  cos  («  —  iW0  --     cosw6>  cosftA/0. 
n  J  n] 

The  original  integral  is  thus  reproduced,  and  after  transpo- 
sition and  division  the  equation  becomes 

f  cos'"#  sin  nd        m     f 

cos"1  8cosn6d6=—  --  1  --     \  cos*-1  6  cos  (n—  i}6dd,  (i) 

J  m  +  n         m+n] 

a  formula  of  reduction  in  which  w  and  n  are  each  diminished 
one  unit. 

If  m  is  a  positive  integer,  we  shall  by  repeated  applications 
of  this  formula  arrive  at  a  directly  integrable  form,  even  when 
n  is  not  an  integer  ;  and  it  may  be  remarked  that,  since 
cos  (—  nff)  =  cos  nd,  the  sign  of  n  may  at  any  time  be  changed 
if  convenient.  As  an  example,  we  have 


cos 


j 


=    cos26>  sin    ^+      cos  B  sin    0+      sin 


75.  If  m  be  negative,  putting  m  =  —  m',  n  =  —  n',  equation 
(I)  becomes 


90  METHODS   OF  INTEGRATION.  [Art.  75. 

fcosn'8  sinn'O  m'      Ccos  (nf  +  i)& 

}  cos«'  0       ~  (m1  +  n')  cosm'e  +  m'  +  n'  }      cos«'+1  0         ' 

in  which  we  may  regard  the  integral  in  the  second  member  as 
the  one  to  be  reduced.  Hence,  putting  n  for  n'  +  I  and  m  for 
m'  +  I,  we  derive 


f 
j 


cos  nQ  sin  (n  —  i)0         m  +  n  —  2  fcos  (n  —  i)6 

^o^e      '    ~w-lcos"'-'6>+       m-i      }      cos-  '0 


If  w  is  an  integer,  we  can  by  this  formula  make  the  integra- 
tion depend  upon  a  final  integral  of  the  form  J  -  ^^6,  which 

is  readily  evaluated  when  n  (and  therefore  />)   is  an  integer, 
and  also  sometimes  when  n  is  a  fraction.     For  example, 


'cos  |0      _         sin  \B       2$  fcos 

"" 


+  -  — 


•dB 

i[cos|0 


2COS20        4L  COS  0  2}  COS 

sin  £0       5  sin  £0        5  fcos  <f>d(j) 

2  cos2  0       4  cos  0        4;    cos  20 

sin  £0        5sin£0          5  i  +  4/2  sin 

_  _i_  ,    i    locx  

2  cos2  0      4  cos  0      8  |/2     s  i  — 


'J 


sn 


Formulae  of  reduction  may  be  deduced  in  like  manner  for 
the  integration  of  cos™  0  sin  n6dd,  and  similar  expressions. 

Reduction  of  Algebraic  Forms. 

76.  The  method  of  reduction  may  be  applied  with  advan- 
tage to  some  algebraic  forms.     Take  for  example  the  form 

xmdx 


§  VI.]  REDUCTION  OF  ALGEBRAIC  FORMS.  9! 

in  which  m  and  n  are  supposed  positive.     Integrating  by  parts, 
taking  u  =  xm  •'  and 

xdx 


we  have 

f     xmdx  xm~*  m  —  i    f     xm~2dx 


(a  +  bx*}"  ~        2b(n  —  i)(a 

a  formula  of  reduction  in  which  m  is  reduced  by  two  units  and 
n  by  one  unit. 

If  m  and  n  are  integers,  the  expression  to  be  integrated  is 
a  rational  fraction  and  includes  (after  a  simple  transformation) 
all  cases  of  multiple  imaginary  roots.  Repeated  applications 
of  the  formula  will  reduce  n  to  unity  or  else  reduce  m  to  zero 
or  to  unity,  the  final  integral  vanishing  in  the  last  case.  For 
example,  the  integral  occurring  in  Art.  23  is  thus  reduced  to 


+  S^x  +  ax*  +  x*  x* 

—  dx~-  - 


a  - 


{     ^dx 

}  (a2  +  x 


2(a*  +  **)  a       2(0*  +  x*}          (a2  + 

in  which  the  final  integral  only  requires  further  transformation  ; 
the  result  is 


x 


The  final  integral  is  also  of  an  integrable  form  when  n  is 
half  of  an  odd  integer. 

General  Method  of  Deriving  a  Formula  of  Reduction. 

77.  The  method  of  integration  by  parts  may  be  said  to 
consist  in  finding  a  function,  uv,  of  which  the  differential  con- 


92  METHODS   OF  INTEGRATION.  [Art.  77. 

tains,  as  one  part,  the  differential  to  be  integrated  ;  so  as  to 
make  the  integration  depend  upon  that  of  the  other  part.  If 
this  last  is  of  the  same  general  form  as  the  given  differential, 
we  have  a  formula  of  reduction. 

The  more  general  method  consists  in  finding  a  function,  P, 
whose  differential  can  be  expressed  as  the  sum  of  certain  mul- 
tiples of  two  or  more  differential  expressions  of  a  given  general 
form.  We  then  have  an  equation  connecting  P  with  two  or 
more  integrals  of  the  given  general  form.  Among  these  we 
select  that  of  the  highest  degree  as  the  one  to  be  reduced,  and 
so  prepare  a  formula  of  reduction  for  the  given  integral  form. 

For  example,  to  find  a  formula  of  reduction  for  the  form 

,  dx  \dx 

or 


J.(#+  2bx -\-  ex'4 

where  X  is  a  quadratic  expression  which  for  convenience  is 
written  in  the  form 


whence  dX  —  2(b  +  cx)dx. 

If  we  take  X~m  for  /*,  we  shall  find  it  impossible  to  reduce 
dP\.Q  the  required  form.  The  same  is  true  if  we  take  xX~m  '; 
but  by  a  combination  of  these  forms  we  are  able  to  accomplish 
our  object.  Thus,  taking 


Xm 

(in  which  p  and  q  are  constants  to  be  determined),  we  have 
qdx       m(p  +  qx]dX  _  qdx  _       (p  +  qx)(b  +  cx}dx 

dr  —      --  —  --  -  2W  —  -  . 

Xn  X  X  X 

The  first  term  is  of  the  proposed  form,  and  the  second  will 


§  VI.]      DEVELOPMENT  OF  AN  INTEGRAL  IN  SERIES.         93 

be  so  if/  and  q  be  so  taken  as  to 'make  (p  +  qx](b  4-  ex]  a  mul- 
tiple of  X  plus  a  constant.  This  will  be  the  case  if  we  put 
q  =  c  and  /  =  b,  which  gives 

-  ex]  =  (b  4-  ex}2  =  cX  —  ac  -f  ^2. 

TJT  U  D  ^    +    CX 

Hence  we  have  /*=- , 


.  „      ^JT  r^f  —  ac  +  £2  , 

and  «/^  = 2w —  #;r 

^"*  ^"" 

/              \dx             ,           ,9x  i^r 
=  dl  —  2w) —  -f  2m(ac  —  o2} , 

\  /  ytn      '  \  I  vm  +  i> 

A.  Ji. 

which  is  in  the  required  form.     Since  the  last  expression  is  of 
the  higher  degree  in  X,  we  put  m  +  I  =  n,  and  obtain 

'dx  _  b  +  ex  (2n  —  3)^        f   dx 


x  _ 
X"~ 


2(n- 


the  formula  of  reduction  required. 

When  n  is  an  integer,  the  final  integral,  in  successive  appli- 
cations of  this  formula,  is  reducible  to  one  of  the  forms  (A),  p. 
18,  or  (£'),  p.  9;  and  when  n  is  half  of  an  odd  integer,  it  dis- 
appears by  virtue  of  the  factor  2n  —  3,  the  last  integration 
being  equivalent  to  the  use  of  formula  (/),  p.  63. 

Development  of  an  Integral  in  Series. 

78.  It  is  often  desirable  to  express  an  integral  in  the  form 
of  a  series  involving  powers  of  the  independent  variable  x, 
especially  in  the  case  of  those  integrals  which  cannot  be  ex- 


94  METHODS   OF  INTEGRATION.  [Art.  78. 

pressed  by  means  of  the  elementary  functions,  that  is  to  say, 
expressions  which  we  now  regard  as  not  integrable. 

The  most  obvious  way  to  do  this  is  to  develop  into  a  series 
the  expression  under  the  integral  sign,  and  then  to  integrate 

te* 

term  by  term.     Given  for  example  \—dx.     Using  the  expan- 

sion of  e*,  we  have 

e*       i  x       x%  xr~'i 

-  =  -  +  i  +  -,  +  -.  +  ...  +  —  r  +  .  .  . 

xx  2  !       3!  r\ 

Hence 


.  ..... 

1 

The  series  integrated  is,  in  this  case,  convergent  for  all 
values  of  x,  from  which  it  readily  follows  that  the  result  of 
integration  is  also  a  convergent  series  for  all  values  of  x. 

If  we  put  • 


/w= 

and  f(x)  can  be  developed  by  Maclaurin's  Theorem,  we  have 

A*)=. 

and  integrating, 


j-y       \  I       /•/       \    _y  /~*       .          r-f  _  \  _^/A»Y"  V//  /      X**1  /      \ 

•*l*)  =    /(^)^  =  ^  +/(o)^  +/x(o)-  ^/''(o)-,  +  . . .,  (i) 

J  21  3! 

in  which  C  =  F(d).     The  result  is  the  same  as  the  develop- 
ment of  F(x)    by  Maclaurin's   Theorem,   since  f(x)  =  F'(x\ 


VI.]  BERNOULLI'S   SERIES.  95 


Bernoulli's  Series. 

79.  By  successive  applications  of  integration  by  parts,  we 
have 


and  finally,  provided  neither  /"(#)  nor  any  of  its  derivatives 
to  the  wth  inclusive  becomes  infinite  for  £  =  o,  z  =  x  or  any 
intermediate  value  of  z, 


\Xf(z)dz  =  xf(x)-^f'( 


3! 

-M  !  ^    ^  ^     I     -i»  t  *  \    / 


which  is  known  as  Bernoulli's  Series.  It  is  not,  in  the  ordi- 
nary sense,  a  development  in  powers  of  x,  because  the  coeffi- 
cients themselves  contain  x\  they  are  in  fact  the  values  of/(;r) 
and  its  derivatives  at  the  upper  limit,  instead  of  their  values  at 
the  lower  limit,  as  in  equation  (i)  of  the  preceding  article. 
The  result  may  in  fact  be  derived  from  that  equation  when 
written  in  the  form 

f  *  x*  x5 

J\Z)CtZ  =  J(Q)X  -\-  j   (Ol— :  -f-  J    (O) — :  T (3) 

Jo  2:  3 • 

For,   putting  z  —  x—  y,  and  denoting  f(x  —  y)  by  <P(y}, 
we  find 


\*f(z}dz  =  -  \f(x  -y)dy  =  f 

•  o  J  x  J 


g6  METHODS   OF  INTEGRATION.  [Art.  79. 

Developing  the  last  integral  by  equation  (3),  we  have 

\"f(z)dz  =  \<t>(y)dy  =  0(o)*  +  0'(o)^  +  0"(o)^  +  .  . ., 

jo  Jo  2:  3 i 

or,  since  <p(y)  =  -  f\x  -  y\    <p"(f)  =  f"(x—y)  . . .,  so  that 
0(0)  =/<»,  0'(o)  -  -/'(*),  0"(o)  =  /"(*)  -  -  , 


which  is  Bernoulli's  Series. 

Taylor 's   Theorem. 

80.  Applying  Bernoulli's  Series,  equation  (2)  of  the  pre- 
ceding article,  to  the  function 


whence 

/'(*)  -  -  F"(x.  +*-*),     /"(*)  =  ^' 
and  using  h  for  the  upper  limit,  we  have 

"(x0}  +  ~ 

3  • 


n  .  j  o  n  . 


But,  denoting  by  F  the  function  of  which  F'  is  the  de- 
rivative, the  value  of  the  first  member  is  F(x0  +  h)  —  (FxJ)  ; 
therefore 


-if  F"+1( 
n  \  J  o 


which  is  Taylor's  Theorem  with  the  remainder  expressed  in 
the  form  of  a  definite  integral. 


VI.]  EXAMPLES.  97 

Examples   VI. 

£•  •  -i       vr  —  nT— -  — 

j  —Jo 

«.     sec-'.tfdJ*,  tfsec-1.*-  —  log  \x  +  y(x*  —  i)]. 

t T  7T         log  2 

3.      tan-1  .#<&?, 

Jo 


4  2 


f  A^+I  r  i 

4.  -r^log.*:^.  — ; —     log* 

J  n  +  i  L  »  +  ij 

71 

5.  29sin 

»  r-. 


f    \z  JL.  Tt.m7tz.nimt 

o.    I    9  cos  #z9  <rat  —  sin ^  sin  — . 

J^  2m          2         rn  4 


i  +  x                      x 
tan"1  x . 

2  2 


»l      "Y    P-*"  /^V*  ^^P&   «     f>'V0'£    -4—     /J0J'     ••-    *> 

•      I    •-*•   C      (/Jt-j  **•  C-        ••   2J*c/       T^    ^C/              «• 

Jo 

f  » 

O(     I  ^f  SCC  ™    ^C  (IOC)  2  I  *^"    SCC ""    ^*  ^~  T  \*^    "" —  ^  /  J  * 

P      .  r^r  1  ,                   /7T       \            /ar  ,    \"1?      JT-/I 

to.    I    6  sm    — h  6    d&.    —6  cos  (  — h6)+sm(-+6)       = 

J0          L4  \4        /            \4       /J0          4 


g  METHODS  OF  INTEGRATION.  [Ex.  VI, 

11.  Lcseca.*d&?,  x  tan  x  +  log  cos  x. 

12.  \xtan.yxdx\  —  \x  (sec3  x — i )  dx    ,  x  tan  x  +  log  cos  x x*. 

15.    \x*  sin  j?  dxt  2x  sin  ^  +  2  cos  x  —  x*  cos  x. 

f  *  """)  *  I* — 

14.    I    .#  sin~  *  .#/&:,  -^"sin-1^ T  si 

•J  o  . — I  o  *  o 


sm'  o       =  - 


xsta.n~1x      x*      log  (i  +  x3) 

—  ft  •     yy.»    4.  n  •*%  —    I     •>*   yV A*  . °       ^ i 

15-     \X  **>  366 

f1  i  2  -\-  x1  ~]I7r2 

16.  I    Ar'sin-1^^,  -  •*•' sin  - J  jc  H /j(\  —  x*}  \    =7- 

Jo  3  9  Jo       6       9 

17.  &"xCO&x4Xm  =  ~  • 

I  22 

*  o  —Jo 

18.  U'^^cos^^c, 


cos     exvint  sin 


f         •  »     ^    I        T  f     ^/  ^^ 

19.  le'^sm  xdx\  =-\e-*(i — cos2x)ax\, 

—  (cos  zx  —  2  sin  2x  —  «;). 
10  v 

^ 

r  ee  if     i 

20.  4eesin6</0,  —(sine  — cose)      =-. 

2  — J0  * 


VI.J  EXAMPLES. 


21.     e*sin  .rcos  xdx.  —  (sin  2x  —  2  cos  2x). 

J  io 

—  sin3  mQ  cos  mQ       36        3  sin  mO  cos  wO 


22. 


[sin4 

Jo 


8  8m 


23.  Derive  a  formula  of  reduction  for    (log  .*)*  xm  dx,  and  deduce 
from  it  the  value  of    (log^r)3^  dx. 

t  xm+1  n       f 

(log  x)n  xm  dx  =  (log  x)n  ---    '•  —    (log  x\n~  *  xm  dx. 
J  m  +  i       m  +  i  j^ 

f/ 

|( 

* 


24. 


.5. 


.     x  cosa  ^r  </a;,  |-^  sin  x  cos  or  —  ^  sin4  x  + 


26.  Derive  a  formula  of  reduction  for  Lt^sin  (x  +  a)  dx,  and  de- 
duce from  it  the  value  of  Lr5  cos  •#<&'. 

\xn  sin  (x  +  a)  dx  =  —  xn  sin  \  x  +  a  H  — 

+  n  Lr""1  sin  \x  +  a  +  —  J  dx. 
Lr6  cos  x  dx  —  (x6  —  2QX*  +  120*)  sin  x  +  ($x*  —  6ox*  4-  120)  cos  x. 


JOO 


METHODS  OF  INTEGRATION.  [Ex.  VL 


f      .      .  4  sm  0  cos  0      sm  0  cos  0        i  r          .    ,         .  7 

27.  cos  9  sin4  0^9, 1-  —  [0  —  sm0cos0J. 

J  6                     24             16 

ir  IT 

f-t  I      f2      •     i  /)/     7/v            3^ 

28.  cos  0  sin  0  </0,  —      sin  u  du  =  -— . 
J0  32J°                        512 


sin0cos30   ,   3sin0cos9 

--  -  +  '  —  - 


30.        cos 


sin0  cos0  (8  cos4  9  +  iocosafr  +  15)  +  150  "b  _  9  4/3  +  i1?^ 

f  cos4  0  cos3  0  _  3  cos  fj  _  3  log  tan  |9 

J  sin3  0     '  2  sin2  02  2 

sin  0          i  .  FTT      0  ~I 

-  — ^ — log  tan     — r  ~ 

8  cos  0       8  |_4       2  _j 


, 

>   J  cos4  Q  3  cos3  9       3  cos  0        2 


cos  01 2          f2  48  —  I57T 

—  os4  9  /*  =  -      — ^-~ 

32 


7T 

f1  ^0  I  [4     *%      _  £ 

'  J0  (i  +  cose)3'  2  Jocos40'      3* 


—     -r, 

sinecosV  3  cos  6       cos  9 


§  VI.]  EXAMPLES. 


101 


f         </9  i  _  3  cos  0        3  e 

37'  J  sin  0  sin2  20 '  4  sin2  0  cos  0       8  sin2  6       8     g     n  ~z  ' 

38.  Prove  that  when  n  is  odd 


sec*-1 6   ,   sec*-»e 

H 1- +  log  tan  Q  ; 


Jsm  0  cos"  0        w  —  i          n  —  3 
and  when  n  is  even 


</e  sec" -'9      sec*-*0  Q 

1 h +  log  tan-  . 

— 


J  sm  0  cos"  0        #  —  i          «  —  3  2 

•  -^  .«  3  *\^ 1 

39.  -r-3- — ,  —  — 7-j- 5-  +  - h  sec  9  +  log  tan  -    . 

J  sine  cos  0         2  sin  9  cos  9       2  {_     3  2j 

> 

40.  .  3 c ,     jP«/  ^  =  sec  0. 

%xy(x*  —  i)  +  ^log  [x  +  y(x*  —  i)]. 


f        •      ^ 

41.   J(«   -jc2)2^, 


(         dx  i   |7       4  „   ,A 

.          -r-j— jr-3,  — H     COSQ^9= 

J0  («     +   *T  «     Jo 

f/    a     ,        n\     //    2         ^S\    j  Sa*    •        ,X    j    •^V'(^—  ^2)[ 

.        (^2  +  ^2)  V(<?  ~  **)  <&,  V  Sm~           +  ~                  "ft 

J  o                 #                                 o 


44. 


«  f  ^V;r 
4S-  J(^  +  i; 


IO2  METHODS  OF  INTEGRATION.  [Ex.  VI. 

f  cos3  6  —  sin's  [,  .     cos  5  — sin  5      ,  ~1 

46-      /  •    ,    ,        —7\idb      =     (i  +  sm  6  cos  6)  ^ -— — 7         — ^  </6    , 
J(sm6  +  cos  6)         L      J  (sin6  +  cos&) 


sin  6  cos  0 
sine  4-  cos0* 


47.  Derive  a  formula  for  the  reduction  of  x  sec"  ^  dx  ;  and  refer- 
ring to  Ex.  n,  thence  show  that  this  is  an  integrable  form  when  n  is 
an  even  integer.  Give  the  result  when  n  —  4. 


,  2^tan  x 

x  sec"  x  dx  = 


n  —  i  (»—  i)(«—  2) 

.     »  —   2    f 

-  Lv  sec*~ 

»—  i  J 

I                         x  sec2  x  tan  .*•      secs  ^      2  r  , 

J  x  sec  A:  dx  = h  -  [*  tan  ^r  +  log  cos  x\. 

48.  Derive   a  formula  of   reduction  for  Lr  cos*  .#<£:,  and  deduce 
-v  cos3  x  dx. 


from  it  the  value  of 


f  AT  cos"-1  AT  sm^r      cos"^"      n—  if 

I*  cos"  .*<&=; h 5 — h-       -Ucos*-2^^. 

J  n  n  n      } 

t  x  sin  x  cos  .r  , 

\x  cos3  ^r  </AT  = —  (cos*  x  +  2)  -\ (cos2  x  +  6). 

Jo  y 

49-   f  cos3  6  cos  40<tf,          '  cos2  fl  sin  4^  +  cos  6  sin  3^      sin  2^ 

J  6  12  24- 


§  VI.]  EXAMPLES.  103 


50    j  cos4 


s4^^  sin  3$        4  cos*  0  sin  2#       12  sin  ^       4  sin* 


7  35  35  35 

i  n         1/1 
5r- 


f       3d         igjg  2sM0f      3a       6cos'0   ,    8         .16! 
1  cos  c7  cos  -ku at>,  cos   u  H -\ —  cos  P  H . 

J  7       L  5  5  5J 


2  sin 
2  sin 


f  cosjfldfl  sinffl      £        (i  +  sin$0)(i  + 

53'  J     cos2^    '  cos^  +6     g(i  -sini#)(i  - 

54.  Derive  a  formula  of  reduction  for  J  cos**  6  sin  nOdti. 

m  _  [cos-  0  sin  (*-,)l^fc 

n  J 


m  +  n  m 


CE;.    I  v'cos  0 sin  ^QdO,  I/cos  0 .  cos  3# (cos  6}^. 

J  T  21 

«;6.  Derive  a  formula  of  reduction  for ZTTT? 

J    cosm  0 

fsin  nO dO  _       cos  (»  —  i)0  m  +  n  —  .2  f  sin  («  —  i}6 d6 

J   cos'"  ^         (w  —  i)  cos'""1  0  w  —  i      j         cos"*"1  6 

dx 
57.   Derive  the  formula  of  reduction  for    -. —         ,,.^  by  the  method 

of  Art.  77,  and  deduce  the  value  of     -. —    — JTT. 

J  (i  —  x) 

P          Jv  •y  vn  —  i      t  ffv 


24(1  -#' 


.! 
g 


IO4  METHODS   OF  INTEGRATION.  [Ex.  VI. 

f       x*dx  x*  x*  x 

5    J  (i  -  *')•  '    10(1  -x'y  ~  16(1  -  *')4  +  32(1  -  x'}3 


___  _  _ 

128(1-**)'       256(1-**)"     512      S  i  -x 


f      x*dx 

>•  J  (t  +  g«y» 


*  * 


>-/        ,        2  \  s     '  /  a  \a 

6(1  +  x  )         24(1  +  x  } 

x  tan" '.a; 

16(1  +  x*)  4        16      ' 

60.  Evaluate  J  -. — '-    — JTJ-,  («)  by  the  formulae  of  reduction  ; 

J    \I      T~    X   f 

(ft}  by  substituting^  =  i  +  x*.    Show  that  the  results  are  identical. 

f      x*dx  x4  x*  i 

(*)      j—    -^t  = T—    —^ 7—    ~^~*  ~  7TT~    ~^\*' 

)(l    +   X  )  IO(l    +  X  )  2O(I    +   X  )  6o(l    +   X  ) 

(0\    f      x*dx      _  i  i i 

f  -y*   //I"  "IT  I  •?  /7      -^—    /I  1*    I 

^v  w^^.  ^i  <t<-  ^.^v    i  r  //a  a\T 

J       «  _     «\i'  »  -     "i*  g  *"*         ^^    ~  a  '* 

j£r 


O  ^-    T                                               ^>  -y»    ^_     T  .<  *>  -V     -4—     T 

£,,&                 1                                               ^Jv                 1  4-  T    *J*        I        1 

I      I  tor*"1  — 

'i                             \1  '         /     2                              \  /       Ldll 


f  _  ^  _ 

63-     7T~  ~Tf> 

J  (^   —  2jc  cos  a  +  ia 


)a 
.r  —  cos  of  2(x  —  cos  a) 


3  sin3  a(x*  —  2X  cos  a  +  i)3    ,3  sin4  ac(x*  —  2X  cos  a  +  iy 


I  F       JC  —   I  I  .         X*  +  X  +    I  _,  2*  +   I~| 

-    -5—     —  ;  ----  H  log  —  :  -  ^  --  1-24/3  tan     -  ;  —    • 
9\_x*  +  x  +  i       x  —  i  (x—1)  V$    J 


VI.]  EXAMPLES.  10$ 


65-   Derive 

f(a  +  bz 

a  formula  of  reduction 
;2)"                    (a  +  bx*}n 

[(a  +  bx'Y^ 

i 

xm 
nb     t(a  +  bx*y-* 

x- 
66.  Derive 

r        dx 

(m  —  i)xm~ 
a  formula  of  reduction 
-  \/(a  +  bx*} 

m 

fr>r 

-  J         x™-* 

J* 

(m  — 

m  y(a  +  bx*}' 

2}b  f                     dX 

\xm  y(a  + 

bx*)        (m  —  i)axm~l 

(m  — 

i}a\xm-2  \f(a  +  bx*}' 

67.  Develop    J       —  dx  in  the  form  of  a  series. 

Jo     x 


sin  x  ,                     xx 
ax  =  x 


5-5!       7-7! 


VII. 

The  Integral  and  its  Limits. 

81.  Before  proceeding  to  some  formulae  of  integration 
involving  special  values  of  the  limits,  it  will  be  convenient  to 
resume  the  consideration  of  the  integral  as  defined  in  the  first 
section. 

In  Art.  3  we  supposed  a  variable  magnitude  of  which  the 
values  depend  upon  some  independent  variable  x  (but  of  which 
the  expression  as  a  function  of  x  is  as  yet  unknown)  to  vary 
simultaneously  with  x,  while  x  passes  at  a  uniform  rate  over  a 
certain  range  of  values.  And  it  is  assumed  that  the  rate  of 
the  function  is  then  expressed  in  terms  of  x  and  its  assumed 
rate;  or,  what  is  the  same  thing,  that  the  relative  rate  of  the 
function  as  compared  with  that  of  x  is  known.  Denoting  tin's 
relative  rate  byf(x],  a  known  function  of  x,  let  F(x)  denote  the 
function  under  consideration  ;  then  by  the  notation  introduced 

F(x}=    f(x}dx..     ......     (I) 


IO6  METHODS   OF  INTEGRATION.  [Art.  8  1 

F(x)  may  now  be  defined  as  the  function  whose  derivative 
isf(x),  or  whose  differential  is  f(x}dx  ;  hence,  from  this  point 
of  view,  integration  is  the  search  for  the  indefinite  integral,  or 
the  inverse  of  the  process  of  finding  the  derivative  of  a  func- 
tion, equation  (i)  implying  nothing  more  than  that 

d\F(x}\=f(x}dx  .......     (2) 

But  because  this  equation  was  found  to  be  insufficient  to  fix 
the  values  of  the  function  F,  we  introduced  in  Art.  4  the  nota- 
tion of  the  subscript  or  lower  limit  ;  so  that,  when  we  write 


(3) 


we  fully  define  the  value  of  F(x)  by  implying  the  additional 
condition  that  F(a)  =  o.  The  integral  thus  modified  is  some- 
times called  a  corrected  integral,  because  the  indefiniteness 
arising  from  the  unknown  constant  of  integration  has  been  re- 
moved. It  remains  "  indefinite  "  in  the  sense  that  it  is  a  func- 
tion of  a  variable  x  to  which  no  special  value  has  been  assigned. 
It  is  to  be  noticed  that,  in  many  applications,  the  constant  is 
determined  (and  the  integral  thus  "  corrected  ")  by  the  condi- 
tion that  some  other  value  (not  zero)  of  F(x)  shall  correspond 
to  a  given  value  of  x.  These  given  simultaneous  values  of  x 
and  F(x]  are  called  the  initial  values. 

82.  As  explained  in  Art.  4,  the  value  of  x  to  which  corre- 
sponds the  required  value  of  the  magnitude  is  known  as  the 
final  value  of  x,  and  is  used  as  the  upper  limit  of  the  integral. 
Denoting  it  by  b,  we  thus  assume  that,  as  the  independent 
variable  x  passes  from  a  to  b,  the  function  F(x)  passes  by  con- 
tinuous variation  from  its  initial  to  its  final  value  ;  and,  when 
this  is  the  case,  we  may  write 

F(a\      ....     (4) 


§  VII.]  THE  DEFINITE  INTEGRAL,  1 07 

in  which  the  function  F  is  defined,  not  necessarily  by  equation 
(3),  which  implies  F(d)  =  O,  but  by  the  more  general  equation 
(i),  or  simply  by  equation  (2). 

We  may  now  define  the  definite  integral  in  equation  (4)  as 
the  increment  received  by  a  variable  whose  differential  is  f(x]  dx 
while  x  passes  from  the  value  a  to  the  value  b  ;  or,  as  the  mag. 

dx 
nitude  generated  at  the  rate  f(x}—r  while  xt  with  the  arbitrary 

dx          .      ,.  , 

rate  -7—,  varies  jrom  a  to  b. 
at 

The  condition  mentioned  above,  that  F(x)  shall  vary  con- 
tinuously, requires  that,  starting  from  some  finite  initial  value 
F(a),  it  shall  not  become  infinite  or  imaginary  for  any  value  of  x 
between  a  and  b.  The  function  F(x)  cannot  become  infinite  or 
imaginary  unless  its  derivative/^)  becomes  infinite  or  imagi- 
nary. Hence  the  condition  will  be  fulfilled  if  f(x)  remains 
real  and  finite  for  all  such  values  of  x.  It  is,  of  course,  also 
assumed  that  there  is  no  ambiguity  about  the  value  of  f(x) 
corresponding  to  any  of  these  values  of  x.  Cases  in  which 
such  ambiguity  might  arise  will  be  considered  later. 

83.  Since  the  rate  of  x  is  arbitrary,  we  may  regard  dx  in 
the  expression  for  the  integral  as  constant,  and  in  that  case  it 
must  be  regarded  as  positive  or  negative  according  as  b  is 
greater  or  less  than  a  ;  in  other  words,  dx  must  have  the  sign 
of  b  —  a.  When  f(x)  does  not  change  sign  for  any  value 
between  a  and  b,  we  can  thus  infer  the  algebraic  sign  of  the 
definite  integral. 

Thus,  if  b  >  a  and  f(x)  is  positive  for  all  values  between 
the  limits,  F(x)  is  an  increasing  function  of  x,  and  x  is  itself 
increasing  ;  therefore  the  definite  integral  denotes  a  positive 
increment.  For  example, 

(""sin  x  , 

dx 

J0     x 

denotes  a  positive  quantity.      Moreover,  since  in  this  case  f(x) 


IOS  METHODS    OF  INTEGRATION.  [Art.  83, 

is  less  than  unity  for  all  values  between  the  limits,  the  integral 
is  obviously  less  than  the  increment  received  by  x,  that  is,  less 
than  TT. 

It  is  evident  from  these  considerations  that  an  interchange 
of  the  limits  changes  the  sign  of  the  integral  ;  thus, 


which  agrees  also  with  equation  (4).    Again,  we  infer  from  the 
same  equation  that 


(*)  dx  =     f(x}dx 

la. 

if  c  is  between  the  limits  a  and  b\  and  this  is  also  true  when  c 
is  outside  of  these  limits,  provided  the  condition  mentioned  in 
the  preceding  article  holds  for  the  entire  range  of  values  of  x 
implied  by  the  several  integrals. 

84.  Returning  now  to  the  corrected  integral,  equation  (3), 
we  see  that  it  is  the  same  thing  as 

J  */(*)  dx, 

in  which  x  stands  at  once  for  the  upper  limit  and  for  the  in- 
dependent variable  which,  in  the  generation  of  the  integral,  is 
conceived  to  vary  from  the  initial  to  the  final  value.  When 
there  is  any  danger  of  confusion  between  these  two  meanings 
of  x,  it  is  well  to  use  some  other  letter  for  the  independent 
variable  ;  thus 

[*/(*)  ** 

J  a 

has  the  same  meaning  as  the  expression  above  ;  and  in  general 
it  must  be  remembered  that  an  integral  is  a  function  of  its 
limits,  and  not  of  the  variable  which  appears  under  the  integral 
sign  unless  the  same  letter  serves  also  to  represent  one  of  the 
limits. 


§  VII.]  THE   GRAPH  OF  AN  INTEGRAL.  IOQ 

Graphic  Representation  of  an  Integral. 

85.  A  geometrical  illustration  was  given  in  Art.  3,  in  which 
f(x]  was  taken  as  the  ordinate  of  a  curve,  and  the  integral  was 
in  consequence  represented  by  an  area.  We  shall  now  employ 
another  illustration,  in  which  the  ordinate  y  represents  the  in- 
tegral itself,  regarded  as  a  function  of  its  upper  limit.  In  other 
words,  we  shall  consider  what  is  called  the  grapJi  of  the  func- 
tion (or  graphic  representation,  employing  rectangular  coordi- 
nates), and  shall  regard  this  curve  as  derived  from  the  expression 
for  the  integral,  and  not  from  the  result  of  any  process  of 
integration.  Putting 


we  have 

f|  =/(*)  =  tan  0,     ......     (2) 

where  0  has  the  same  meaning  as  in  Diff.  Calc.,  Art.  38. 

Thus  for  every  value  of  x  we  know  the  inclination  0of  the 
curve  to  the  axis  of  x.  The  notation  also  implies  that  when 
x  =  a,  y  —  o.  Starting,  then,  from  the  position  (a,  o),  if  we 
imagine  the  point  (x,  y)  always  to  move  in  the  proper  direc- 
tion, a  direction  which  changes  as  x  changes,  it  will  trace  out 
a  definite  curve,  so  that  the  function  y  has  a  definite  value  for 
each  value  of  x. 

86.  As  an  example,  let  us  take  the  function 


y 


f  sin  x  ' 

=         -Tdx> (0 

Jo        •* 


a  case  in  which  the  indefinite  integration  cannot  be  performed. 
Starting  from  the  origin,  the  describing  point  must  move  in  the 
varying  direction  defined  by 

sin  x  .  . 

tan  0  =  (2) 


110  METHODS   OF  INTEGRATION.  [Art.  86. 

When   x  —  O,  we  have   tan  0  —  I,  or   0  =  45°  ;  the  curve 
therefore  starts  from  the  origin  at  this  inclination.     But,  since 

equation  (2)  shows  that  tan  0 
decreases  as  x  increases,  the 
curve  as  we  proceed  toward  the 
right  lies  below  the  tangent 
at  the  origin,  as  in  Fig.  3. 
The  ordinate  y  continues,  how- 

/i  ever,  to  increase  (compare  Art. 

83)  until,  at  the  point   A,   x 
FlG-  3-  reaches  the  value  7t,  for  which 

0  =  O.  As  x  passes  through  this  value,  0  changes  sign  ;  there- 
fore^ has  reached  a  maximum  value.  From  x  =  n  to  x  =  2n 
tan  0  is  negative,  hence  y  decreases ;  in  this  interval  sin  x  in 
equation  (2)  goes  through  numerically  the  same  values  as  before, 
but  the  denominator  x  is  now  much  larger  than  before,  hence 
it  is  plain  that  y  does  not  decrease  to  zero.  In  like  manner  it  is 
obvious  that  it  will  increase  again  from  x  =  2n  to  x  =  $7t,  and 
so  on,  reaching  alternate  maxima  and  minima,  which  contin- 
ually approach  an  asymptotic  value  of  y  corresponding  to 
x  =.  oo  .  The  form  of  the  curve  for  positive  values  of  x  is  there- 
fore that  represented  in  Fig.  3.  For  negative  values  of  x  the 
curve  has  a  similar  branch  in  the  third  quadrant.  It  is  evident 
that  we  can  in  no  case  have  a  finite  value  of  the  integral  when 
x  =  oo ,  unless,  as  in  this  case,  the  quantity  under  the  integral 
sign  approaches  zero  as  a  limit  when  x  increases  without  limit. 
87.  When  the  graph  or  curve  of  the  indefinite  integral  is 
drawn,  the  definite  integral  between  any  limits  c  and  d  is  rep- 
resented by  the  increment  of  y  in  passing  from  the  abscissa  c 
to  the  abscissa  d,  and  the  condition  given  in  Art.  82  requires 
that  the  curve  shall  be  continuous  between  the  points  corre- 
sponding to  the  limits  ;  in  other  words,  that  these  points  should 
belong  to  the  same  branch  of  the  curve. 

In  the  illustration  above,  f(x)  remains  real  and  finite  for  all 


VII.] 


THE    GRAPH  OF  AN  INTEGRAL. 


Ill 


values  of  x  ;  accordingly,  in  this  case,  all  values  of  the  limits 
are  possible.  But,  if  f(x]  becomes  infinite  or  imaginary  for 
any  value  of  x  between  the  limits,  it  will  usually  be  found  that, 
although  the  indefinite  integral  y  —  F(x)  may  have  a  finite 
value  at  each  limit,  the  corresponding  points  will  belong  to 
separate  branches  of  the  curve,  and  therefore  equation  (4)  of 
Art.  82  cannot  be  used. 

For  example,  if  f(x)  =  x~*,  which  is  infinite  for  x  •=  o,  we 
have  for  the  indefinite  integral 

tdx         -   I 

y  —  —  —  — ' 

J  x  x 

and  y  is  also  infinite  when  x  =  o. 

The  graph  of  this  integral  is  an  equilateral  hyperbola,  by 
means  of  which  we  can  represent,  for  example,  the  definite  in- 
tegral between  the  limits 
—  2  and  —  i  as  the  differ- 
ence BR  between  two  values 
of  the  ordinate.  But  we 
cannot  interpret  in  this  case 
an  integral  between  whose 
limits  the  value  x  =  O  lies 
(that  is,  with  one  negative 
and  one  positive  limit), 
because  the  corresponding 
points  would  lie  on  dis- 
connected branches  of  the 
curve. 

It  will  be  noticed  that, 
for  the  same  reason,  if  we 
use  the  notation  of  the  corrected  integral  we  cannot  represent 
the  entire  curve  by  a  single  equation  ;  thus  the  two  branches 
as  drawn  in  Fig.  4  have  the  separate  equations 
r    dx  (xdx 

y  =  I   ^~  and  y =  \  IF* 

J  -oo  X  J  oo  •*• 


Lff 


FIG.  4. 


112 


METHODS   OF  INTEGRATION, 


[Art.  87. 


the   latter  expression   denoting  an   essentially  negative  quan- 
tity. 

88.  It  may  happen,  however,  that  the  indefinite  integral 
remains  finite  for  a  value  of  x  which  makes  f(x)  infinite.  In 
such  a  case,  if  the  value  of  f(x)  remains  real  while  x  passes 
through  this  value,  we  shall  still  have  a  continuous  curve  and 
a  continuous  variation  of  the  ordinate  between  limits  which 

include  this  value  of  x.  For 
example,  when /"(;«:)  =  x~%,  which 
is  infinite  for  x  =  o,  and  is  real 
for  both  positive  and  negative 
values,  we  have 


y 


fdx 
}^~~ 


FIG.  5. 


The  curve  which  is  drawn  in 
Fig.  5  touches  the  axis  of  y  and 
is  real  on  both  sides  of  it,  form- 
ing a  cusp.  Thus  the  ordinate 
varies  continuously  while  x  passes 


through  zero,  and  we  can  write,  for  instance, 


in  accordance  with  equation  (4),  Art.  82,  as  illustrated  in  Fig. 
5  by  the  difference  of  ordinates  BR. 


multiple-valued  Integrals. 

89.  When  the  indefinite  integral  is  a  many-valued  function, 
the  condition  that  it  shall  vary  continuously  while  x  passes 
from  the  lower  to  the  upper  limit  requires  the  selection  of 
properly  corresponding  values  at  the  two  limits.  In  illustra- 


§  VII.] 


MUL  TIPLE-  VA L  UED   INTEGRALS. 


tion  let  us  construct  the  graph  of  the  fundamental  integral 


=  sn"  *• 


0) 


In  this  case  x  can  only  vary  between  the  values  —  i  and 
-+-  i.  Proceeding  from  the  origin,  the  curve,  Fig.  6,.  has  the 
inclination  45°,  which  is  gradually  in- 
creased to  90°  when  x  =  i,  at  the  point 
A,  for  which  the  value  of  y  is  £TT.  hsx 
passes  from  o  to  —  i  a  similar  branch 
is  described  in  the  third  quadrant,  com- 
pleting the  curve  drawn  in  full  line,  the 
ordinate  of  which  is  in  fact  the  primary 
value  of  the  indefinite  integral  sin'1^- 
(Diff.  Calc.,  Art.  74).  .  But  the  complete 
curve  _y=sin~I..r,  or;r=sin^/,  has  another 
branch  continuous  with  this  at  the  point 
(i,  £?r),  as  represented  by  the  dotted 
line.  In  order  that  the  point  moving 
in  accordance  with  the  integral  expres- 
sion shall  describe  this  branch,  we  must 
suppose  that  x,  after  reaching  the  value 
unity  at  the  point  A,  begins  to  decrease, 
and  that,  as  dx  thus  changes  sign,  the  radical  |/(i  —  x*},  which 
then  becomes  zero,  also  changes  sign,  so  that  the  quantity 
under  the  integral  sign  remains  positive,  and  y  goes  on  in- 
creasing. Since  the  radical  cannot  change  sign,  while  varying 
continuously,  except  when  it  passes  through  the  value  zero  (at 
which  time  its  two  values  are  equal),  it  is  only  when  x  =  ±  i 
that  it  can  return  to  its  previous  values  without  causing  y  also 
to  return  to  the  corresponding  previous  values;  that  is,  with- 
out making  the  generating  point  return  upon  the  path  already 
described. 

We  may  thus,  by  the  alternate  increase  and  decrease  of  x 
between   its  extreme  values,  describe  an  infinite  number  of 


FIG.  6. 


1 14  METHODS   OF  INTEGRA  TION.  [Art.  89. 

branches.  In  this  generation  of  the  integral  the  value  of  the 
radical  undergoes  periodic  change  of  sign  but  is  never  am- 
biguous ;  for,  in  drawing  the  curve  we  assumed  that  in  equa- 
tion (i)  the  radical  had  its  positive  value  when  x  =  o  and 
y  =  o.  This  assumption  determines  the  value  of  the  radical 
for  every  other  value  of  y  ;  it  is,  in  fact,  always  equal  to  cos,  y. 

90.  In  the  direct  application  of  limits  to  the  integral  (i),  it 
is  of  course  sufficient  to  limit  the  indefinite  integral  to  its 
primary  value,  the  radical  being  positive  for  all  values  of  x  be- 
tween the  limits  ;  but,  when  the  integral  is  the  result  of  trans- 
formation, care  may  become  necessary  in  the  selection  of  the 
values  at  the  limits. 

Consider,  for  example,  the  integral 

f*"       dz 

J  _,   |/(2  -  O' 

which  can  be  evaluated  directly  by  formula  (/')  in  the  form 

z  -i  v*      ^n 
sin"1  —        =  — .     But  suppose  we  make  the  transformation 

z*  =  i  —  x,   whence   dz  = ,  and     \/(2  —  z*)  =  ^/(i  +  x}. 

It  is  first  to  be  noticed  that,  because  the  value  of  z  is  neg- 
ative at  the  lower  limit,  we  should  in  the  value  of  dz  put 

dx 

z  =  —  |/(i  —  x}\  thus  dz  =  — —. :,  and  the  result  of  trans- 

£    w  I  1    ^^         ) 

formation  is 

f2       dz  i  f~:        dx 


z         _  i  f~:          x 

_*•>-"  2  Jo  ~W=~?i 


The  relation  z*  =  i  —  x  shows  that  as  z  increases  from  —  I 
to  o,  x  increases  from  o  to  i,  and  the  corrected  integral  in  the 
second  member  (which  is  sin"1  x}  increases  from  o  to  %n. 
Then  as  z  further  increases  from  o  to  -j/2,  x  decreases  from  I 
to  —  I  ;  but  the  integral  continues  to  increase  from  \n  through 


§VIL] 


MUL  TIPLE-  VA  L  UED    INTEGRA  LS. 


values  which  are  not  "  primary,"  reaching  the  final  value  f  TT  at 
the  upper  limit,  so  that  we  have  for  the  value  of  the  second 
member  of  the  equation  ^(f?r  —  o)  =  f  n  as  before. 

91.  To  illustrate  another  mode  in  which  an  integral  may 
acquire  multiple-values,  let  us  consider  the  graph  of  the  fun- 
damental integral 

dx 
— 5-  =  tan  '  x. 

Proceeding  from  the  origin,  at  the  inclination  45°,  the  curve 
approaches,  as  x  increases  without  limit,  an  asymptote  parallel 
to  the  axis  of  x  at  the  distance 
£;r.  The  similar  branch  de-  _ 
scribed  for  negative  values  of 
x  completes  the  full  line  in 
Fig.  7,  representing  the  pri-  _ 
mary  values  of  tan"1  x ,  which 
are  employed  in  any  direct 
application  of  limits  to  the 
integral.  But  the  complete 
curve  y  =  tan"1  x  consists  of 
an  unlimited  number  of  branches  which  are  repetitions  of  this 
curve  at  successive  vertical  intervals  each  equal  to  TT,  so  that 
each  asymptote  is  approached  by  two  branches. 

Now,  when  the  integral  arises  from  transformation,  it  may 
happen  that  x  passes  through  infinity,  changing  sign,  and  then 
the  ordinate  will  pass  without  discontinuity  of  value  from  one 
branch  to  the  next.  For  example,  given  the  integral 


FIG.  7. 


dB 


J0  cos'  6  +  9  sin2  B 
if  we  put  tan  B  =  x,  this  becomes 


_  re     secj 
Jo  i  +.9 


9  tan8  0' 


4/3 
3 


dx      _  i 

—  -  tan- 

3 


=  -  [tan-1  (—  1/3)  —  tan-1  o]. 

0 


Il6  METHODS   OF  INTEGRATION.  [Art.  91. 

As  6  passes  from  o  to  %n,  x  increases  from  o  to  -f  co  ,  and 
tan"1  ^x  increases  from  O  to  £TT.  But  as  6  further  increases 
from  I-TT  to  £?r,  x  changes  sign  and  passes  from  —  co  to 
—  \  1/3.  During  this  interval  tan"1  ^x  still  further  increases 
from  %-n  to  f  re,  which  is  therefore  in  this  case  to  be  taken  as 
the  value  of  tan"1  (—  1/3),  while  we  take  zero  as  the  value  of 
tan'1  o.  Hence 


r?        do 


cos*  6  +  9  sin1  6       9 


Formula  of  Reduction  for  Definite  Integrals. 

92.  The  limits  of  a  definite  integral  are  very  often  such  as 
to  simplify  materially  the  formula  of  reduction  appropriate  to 
it.  For  example,  to  reduce 


xne-*dx, 

}  O 


we  have  by  the  method  of  parts 
\xne-*dx  —  —  xne~ 


\ 


Now,  supposing  n  positive,  the  quantity  xne~x  vanishes  when 
x  =  o,  and  also  when  x  =  oo.  [See  Diff.  Calc.,  Art.  159.] 
Hence,  applying  the  limits  o  and  oo  , 


I   x"e~*dx  =  n\ 

Jo  J 


By  successive  application  of  this  formula  we  have,  when  «  is 
an  integer, 


§  VII.]  FORMULA.    OF  REDUCTION. 


•2-1  =  n  \ 


I   xne~xdx  =  n(n  —  i)  ..... 

Jo 

93.  From  equation  (i),  Art.  66,  supposing  m  >  I,  we  have 

V  Tt 

f2  sin-  OdO  =  ^Hl  f2  sin*-'  Odd. 
J0  m     J0 

If  w  is  an  integer,  we  shall,  by  successive  application  of  this 

»r  JT 

formula,  finally  arrive  at  \  dO  =  -  or     *  sin  0^0  =  i,  according 

J0  2  Jo 

as  m  is  even  or  odd.     Hence 

ir 
•r         •  f  ^     •    »«  a  ja        (m  ~  l}(m  —  3)  •  •  •  •  I      ^  /  r>v 

if  »?  is  even,          smw  6  dO  =  v  --  7—^  -  r-^  ---  ,  .  .  (P) 
J0  m(m  —  2)  ......  22 

* 

j  -r         •        Jj  f  2     •    «.  /i  jn        (m  —  l)(m  •—«)....  2 

and  if  w  is  odd,          sin"*  Q  dQ  =.  *  --  7-^  -  ^^  —  — 
J0  m(m-2)  ......  I 

94-.  From  equations  (3)  and  (4)  Art.  69,  we  derive 


f2  sinw  6  cos"  0^0=   n  ~~  l  (2  sinw  6  cos"-2 
Jo  m  +  n  }0 

w  w 

and         F  sin*  0  cos"  0  </0  =  W~  T   [  2  sin'"-2  0  cos"  0  dB. 

m  +  n  J0 


Il8  METHODS   OF  INTEGRATION.  [Art,  94. 

By  successive  applications  of  the  first  of  these  formulae,  we 
produce  a  series  of  factors  in  the  numerator  decreasing  by  in- 
tervals of  two  units  from  n  —  i,  and  ending  with  i  or  2,  as  in 
formulae  (P)  and  (P').  By  the  second  formula  we  then  pro- 
duce a  like  series  of  factors  in  the  numerator  beginning  with 
m  —  I.  The  successive  factors  in  the  denominator  produced 
in  the  process  form  a  single  series  beginning  with  m  +  n  and 
decreasing  by  intervals  of  2,  the  final  factor  being  greater  by 
2  than  the  sum  of  the  exponents  in  the  final  integral.  Now 
this  final  integral  will  take  one  of  the  four  forms 

ir  it  IT  it 

V  dS,          [2sin  0  dO,         (*  cos  6  dB,         or          f "sin  B  cos  B  dB, 

Jo  Jo  -Jo  Jo 

according  as  m  and  n  are  even  or  odd.     Thus  the  final  factors 
in  the  denominator  are,  in  the  four  cases  respectively, 

2,  3*  3.  4; 

-while  the  values  of  the  final  integrals  are  respectively 


The  result  will  therefore  be  the  same  if  we  carry  the  series 
of  factors  in  the  denominator  in  all  cases  to  I  or  2,  and  ignore 
the  final  integral  except  in  the  first  case  (m  and  n  both  even), 
when  its  value  is  \rc.  Therefore  we  may  write,  when  m  and  n 
are  positive  integers, 

flin-  0  cos"  6  M  =  («-'X"--3)...(»-0(»-3)...gt .  (0 
J0  (m  +  n)(m  +  n  —  2) 

provided  that  each  series  of  factors  is  carried  to  2  or  i,  and  a  is 
taken  equal  to  unity,  except  when  m  and  n  are  both  even,  in  which 
case  of  =.  \ft. 

This  formula,  of  course,  includes  formulas  (P)  and  (P'). 


§  VII.]        CHANGE    OF  INDEPENDENT    VARIABLE. 


Change  of  Independent  Variable  in  a  Definite  Integral. 

95.  A  change  of  independent  variable  is  frequently  made 
for  the  purpose  of  simplifying  the  limits,  rather  than  of  chang- 
ing the  form  of  the  integral.  For  example,  suppose  the  limits 
in  formula  (0  to  be  multiples  of  %n  by  any  consecutive  integers 
k  and  k  -+-  i.  Putting  Q  =  \kn  +  0,  we  have,  according  as  k 
is  even  or  odd, 

(*  +  1)  -  *- 

"1  " 


1  -  *- 

2  sin"  B  cos"  6dB  =  ±  \   sin"1  0  cos 
J*?  Jo 


or 


=  ±  J 


2  sinw  6  cosw  6  dB  =  +  T  cos**  0  sin* 


But  by  formula  (<2)  the  integrals  in  the  second  members 
have  the  same  value  ;  hence  the  numerical  value  of  the  integral 
of  formula  (Q)  is  the  same  for  every  quadrant.  The  sign  to  be 
used  is  obviously  that  of  sin*"  6  cos*  6  in  the  quadrant  in  ques- 
tion. We  can  thus  readily  determine  the  value  of  the  integral 
when  the  limits  are  any  multiples  of  \n.  When  m  and  n  are  both 
even,  the  sign  is  always  -=}-,  and  we  have  to  multiply  the  result 
of  formula  (Q)  by  the  number  of  quadrants.  In  other  cases,  we 
have  two  positive  and  two  negative  results  while  6  describes 
one  revolution.  / 

96.  When  the  limits  of  an  integral  of  the  form  considered 
above  are  o  and  JTT,  we  may  by  introducing  the  double  angle 
obtain  the  limits  o  and  \n.  Since,  if  0  =  26, 

sin  6  cos  0=%  sin  0,     sin"  #=£(i—  cos  0),     cos2  ^=|-(i  +  cos  0), 

the  integral  will,  whenever  m  and  n  are  either  both  even  or  both 
odd,  be  thus  converted  into  one  or  more  integrals  of  the  same 


I2O  METHODS    OF  INTEGRATIONS.  [Art.  96. 

form,  with  integral  values  of  the  exponents,  and  having  limits 
of  the  desired  form. 

For  example,  by  this  transformation  we  obtain 

ir  T 

f  4  sin'  0  cos4  0  dB  =  ~  f2  sin4  0  (i  -  cos  0)  d$ 

J  o  4  J  o 

-  j_r  3-  *  *L  _  3-  i  .  1  1   _j_r  _*_  _  in 

64L4-22       5-3-iJ      64l_3i6       5  J  • 

97.  A  transformation  which  does  not  change,  or  which 
interchanges,  the  limits  may  sometimes  reproduce  the  given 
integral  together  with  known  integrals,  and  thus  lead  to  its 
evaluation.  When  the  limits  a  and  b  of  any  integral  are  finite, 
they  will  be  reversed  by  the  substitution 

x  =  a  +  b  —  z. 
Thus 

f(x]dx  =  -•  £/(a  -M-*)<fe==  I  f(a  +  b-.z}dz; 

or,  since  it  is  indifferent  whether  we  write  z  or  x  for  the  in- 
dependent variable  in  a  definite  integral, 


=   { 

»  < 


+  b-  x)  dx. 

»  <z 

For  example, 
['0  sin4  040=  f  V  -  61)  sin4  (a  -6)48=  ["(*  -  0)  sin4  0  </0  ; 

J  o  Jo  Jo 

hence 

2\'Bs\i-\4Bde=  n  ['sin4  6  d6, 

Jo  Jo 

and,  by  formula  (P\ 


4-22 


VII.]    THE  INTEGRAL   AS    THE  LIMIT  OF  A    SUM.  121 


98.  When  the  limits  are  o  and  oo  ,  they  will  be  reversed  if 
we  assume  for  the  new  variable  the  reciprocal,  or  a  multiple  of 
the  reciprocal,  of  x.  For  example,  if  in 


«=I 

log 

3    _l_ 

a 

x     , 
3  ax 

*dy 

ve  find 

*r  ^ 

y 

p  2  log#  —  log  y 

y* 

>    v 
log 

^ 

hence 

Jo                /+*' 

,00 

11  —   I  nor  /r    I 

ay 

dy 

«  I         a 

Jo/ 

log« 

+  ' 

2a 


The  Integral' regarded  as  the  Limit  of  a  Sum. 

99.  We  have  seen  in  Art.  3  that,  if  the  curve  y  =f(x~)  be 

f& 

constructed,  the  integral      f(x)dx  (in  which  we  suppose/^) 

J  a 

to  remain  finite  and  continuous  as 
x  varies  from  a  to  b]  is  rep- 
resented by  the  area  included  by 
the  curve,  the  axis  of  x  and  the 
ordinates  corresponding  to  x  =  a 
and  x  =  b. 

Let  CD  in  Fig.  8  be  the  curve, 
and  let  the  base  of  this  area,  AB, 
whose  length  is  b  —  <z,  be  divided 
into  n  equal  parts.  Denote  the 
length  of  each  part  by  Ax,  so  that 


FIG.  8. 


b  =  a  +  n  Ax, 
and  erect  an  ordinate  as  in  the  figure  at  each  point  of  division. 


122  METHODS   OF  INTEGRATION.  [Art.  99. 

The  whole  area  is  thus  divided  into  n  parts,  as  represented  by 
the  equation 

rb  fa  +  A*  t  a+  2A*  fb 

ydx  —  ydx  +  ydx  +  .  .  .  -f  ydx.      (i) 

J  a  J  a  J  a  +  A*  J  A-A* 

Now  if  we  denote  the  values  of  y  corresponding  to  a  -f  Ax, 
a  +  2  Ax,  .  .  .  b  by  yl  ,  y^  ,  ,  .  .  yn  ,  the  rectangles  y^Ax,  y^Ax,  .  .  . 
ynAx,  which  are  constructed  in  the  figure,  are  approximations 
to  the  several  terms  of  the  second  member  of  equation  (i). 
Hence  the  sum  of  these  rectangles, 

2-yAx  =  y,Ax  +  y^Ax  +  .  .  .  +  ynAx,   ...     (2) 


is  an  approximate  expression  for  the  integral  in  equation 
(i).  When,  as  in  Fig.  8,  y  increases  continuously  while  x  passes 
from  a  to  b,  the  sum  of  the  rectangles  exceeds  the  curvilinear 
area,  that  is,  2  yAx  exceeds  the  integral. 

If  we  take  for  the  altitude  of  each  rectangle  the  initial 
instead  of  the  final  value  of  y  in  the  interval,  we  shall  obtain  in 
like  manner  an  approximate  expression, 

2'  yAx  =  y^Ax  +  y^Ax  +.....'+  yn-*  Ax,       .     (3) 

which  will,  in  this  case,  represent  an  area  less  than  the  curvi- 
linear one,  so  that  2'yAx  is  less  than  the  integral.  Thus  the 
integral  is  intermediate  in  value  to  the  two  expressions  in 
equations  (2)  and  (3),  of  which  the  difference  ib 

2  yAx  -  2  'yAx  ==(>«-  y0)Ax.      ...     (4) 

Therefore  the  difference  between  the  integral  and  either  of 
the  approximate  expressions  is  less  than  (yn  —  y0)  Ax. 

Now  when  the  number  of  parts  n  is  indefinitely  increased, 


§  VII.]    THE  INTEGRAL   AS    THE  LIMIT   OF  A    SUM.  123 

so  that  Ax  is  decreased  without  limit,  the  limit  of  (yn  —  j0)  Ax 
is  zero  ;  it  follows  that  I  ydx  is  the  limit  of  2y  Ax  for  any 

range  of  values  of  x  for  which  y  is  an  increasing  function  of  x. 
It  can  be  shown  in  like  manner  that  the  same  thing  is  true 
while  y  is  a  decreasing  function,  and  therefore  in  general 

[  f(x]dx  —  the  limit  of^/(*)  Ax    .     .     .     (5) 

J  a  x  =  a 


when  Ax  decreases  without  limit. 

100.  The  typical  term/(;r)  Ax  is  called  the  element  of  the  sum 
in  equation  (5).  It  will  be  noticed  that  the  sum  will  have  the 
same  limit  whether  the  x  in  f  (x)  Ax  be  taken  as  in  equation 
(2)  or  as  in  equation  (3),  or  in  any  other  manner,  provided  it  be 
some  value  within  the  interval  to  which  Ax  corresponds.  The 
expression  f(x]dx  is,  in  like  manner,  called  the  element  of  the 
integral  in  equation  (5). 

In  many  applications  of  the  Integral  Calculus,  the  expres- 
sion to  be  integrated  is  obtained  by  regarding  it  as  an  element. 
In  so  doing  we  are  really  dealing  with  the  element  of  the  sum  ; 
but  as  we  intend  to  pass  to  the  limit  we  may,  in  accordance 
with  the  remark  made  above,  ignore  the  distinction  between 
any  values  of  x  between  x  and  x  +  Ax.  By  equation  (5),  we 
pass  to  the  limit  by  simply  writing  d  in  place  of  A,  and  the 
integral  sign  in  place  of  that  of  summation,*  so  that  in  practice 
it  is-  customary  to  form  the  element  of  the  integral  at  once  by 
writing  d  instead  of  A. 

*  The  term  integral,  and  the  use  of  the  long  s,  the  initial  of  the  word  sum, 
as  the  sign  of  integration,  have  their  origin  in  this  connection  between  the 
processes  of  integration  and  summation. 


124      ADDITIONAL   FORMULAE   OF  INTEGRATION.  [Art.  IOI. 


Additional  Formula  of  Integration. 

101.  The  formulae  recapitulated  below  are  useful  in  evalu- 
ating other  integrals.  (A)  and  (A')  are  demonstrated  in 
Art.  17;  (B)  and  (C)  in  Art.  29;  (D)  and  (E)  in  Art.  30; 
(F)  in  Art.  31  ;  (G)  and  (G1)  in  Art.  35  ;  (H)  and  (/)  in  Art.  50  ; 
(/)  in  Art.  51  ;  (K)  in  Art.  52  ;  (L)  in  Art.  53  ;  (M)  in  Art.  55  ; 
(N)  and  (O)  in  Art.  58  ;  (P)  and  (/>')  in  Art.  93  ;  and  (0  in 
Art.  94. 

dx  \  x  —  a 


t     dx  \          x  —  a      \     dx  I  #  +  .2- 

-a  --  5-  ==  —  log-      -  ;     -5  --  2  =—  log-     —  .  .     .    (A') 
}x*  —  a?         20.     *>  x  +  a     \ai  —  xi        2.a     *  a  —  x 

(sw?OdO  =  \(d  -  sin  0  cos  0)  ...........   (B) 

{cos*edd  =  %(e  +  sin0cos8)  ...........  (C) 

(          df) 

—  2j=  log  tan  0  ...........    .'.(/?) 

J  sin  B  cos  8 


i  —  cos  0 


(  dV  Fit     0~\  i+sin0 

—  s=logtan    -  +  -    =log  -     -  =  log  (sec  0  +  tan  0).    (F) 
JcosO  [_4    2J  cos^ 

^—j,  -  a  =  ~77~F  —  m  tan"'  \V  ^-TT  tan 

J(7  +  <?COS^  -/(a2  —  ^)  L        «  +  ^ 


§  VII.]      ADDITIONAL   FORMULA   OF  INTEGRATION.         125 

d9  i 

-^\{°S 


a)  —  V(b  —  a]  tan  \  6 ' 
+  a*)-a 


dx  i         a  — 


x 


2 


f 
J 


ft-  a 


;  log  \*  +  V(^  ±  a")]  .    . 


sin-  OdS  =  f'  cos-  Odd  =  (**-00»-3)---i  ^ 
Je  mm  —  2  ......  22 


126  METHODS   OF  INTEGRATION.  [Art.   IOI. 


psin-*<0=  Fcos^^  =  ^-/)(^--3)---2t 
Jo  Jo  m(m-2)  ......  i 

(7  .  .        (m  —  i}(m—3)--x(n  —  i)(n—'$)--'  /XYV 

sin™  6  cos*  6dO  =  *—.  -  ,A  w   J/,  -  v    x    A  -  «  --  «      .  (Q\ 
}  (m  +  n)(m  +  n  —  2)  .......... 


in  which  a  =  i,  unless  /«  and  w  are  both  even,  when  «  =  — . 

2 


1 

Examples  VII. 


fwir  .Jh 

J0  JT^Te'       [a > b* and  w  an  integer] 


>.   f'sin4 

Jo 


tznir±.  2          ^TQ 
?"    Jo  2    +  COS  9* 

•a 

j.    f2  sir'  «"fo 

Jo 

sh 

Jo 


3.  i    a^,  32 


16 

4.    |  sm"  ficfy,  — 


16 

35' 


Qcos  6^r9,  S12 


VII.]  EXAMPLES,  127 

fff  4 

7.  sin3  0  cos4  0</0.  — . 

J°  35 

*  it 

f»  •  i  I"?  • 

8.  sinw  0  cosw  9  </9,  sin™  0  </0 . 
J°                                                 ry                          2WJ0 

f1      x™dx  1-3-5   •   '   '   (2«  — i)  5 

2-4-6    •     •     •     •     2«        2 
2-4-6-     •     •     •   2« 


f1     ^2"+I  <& 

10-      4/fi-^r 

jo    y  \A       •*  / 


3-5-7-   -   - 


II*  I  *^«      \  W  fc^       /  w*^r. 

63 


(^-a^dx 

12.     '         V 


'-I 

-I: 


. 
160* 

8 


r      x*dx  n 

14-  Jo  (a*  +  x')f'  J^' 

15.  Prove  that 

I    aJ*"'^ — x)m"T  dx  =  J    ^r**'1  (as— Ar)**"11  ^£r, 

Jo  Jo 

and  derive  a  formula  of  reduction  for  this  integral,  supposing  n  >  o 
and  »z  >  i. 


w j  f« 

= x»  (a— 

n    J0 


128  METHODS   OF  INTEGRATION.  [Ex.  VII. 

16.  Deduce  from  the  result  of  Ex.   15   the  value  of  the  integral 
when  m  is  an  integer. 


-'    a-x-x=      .  .a"*-'. 

n  (n+i)   •   •    -  (n  +  m—i) 

ta  4  218  4/2     JL& 

17.  (a  +  x}"  (a-xy  dx.     See  Ex.  16.  —a8 
J-a                                                                                45°4S 

IT 

18.  Tsui'  6  (cos  0)2  ^/9.     /'a/  sina  6  —  x,  and  see  Ex.  16. 

Jo 


w 

19.  I4 cos8  0</0, 

J  o 


5-7-H-I9 

* 

•?^7T  < 

«J*J  I        J 

256      24 


20.  ,  . 

384 

371"  ~~  4 

" 


22.  Evaluate      0  cos1  0a$  (see    Ex.  VI,  48)    and   thence   derive 

J  o 

I  0'  cos*  0  dti  by  the  method  of  Art.  97. 

J  o 


I47T 

9 


§  VIII.]  PLANE  AREAS. 


CHAPTER   III. 

GEOMETRICAL  APPLICATIONS— DOUBLE  AND  TRIPLE 
INTEGRALS. 


VIII. 

Plane   Areas. 

102.  THE  first  step  in  making  an  application  of  the  Inte- 
gral Calculus  is  to  express  the  required  magnitude  in  the  form 
of  an  integral.  In  the  geometrical  applications,  the  magni- 
tude is  regarded  as  generated  while  some  selected  independ- 
ent variable  undergoes  a  given  change  of  value.  The  inde- 
pendent variable  is  usually  a  straight  line  or  an  angle,  varying 
between  known  limits;  the  required  magnitude  is  either  a 
line  regarded  as  generated  by  the  motion  of  a  point,  an  area 
generated  by  the  motion  of  a  line,  or  a  solid  generated  by  the 
motion  of  an  area.  A  plane  area  may  be  generated  by  the 
motion  of  a  straight  line,  generally  of  variable  length,  the 
method  selected  depending  upon  the  mode  in  which  the 
boundaries  of  the  area  are  defined. 

The  Area  generated  by  a  Variable  Line  having  a  Fixed 

Direction. 

103.  The  differential  of  the  area  generated  by  the  ordinate 
of  a  curve,  whose  equation  is  given  in  rectangular  coordinates, 
has  been  derived  in  Art.  3.  The  same  method  may  be  em- 
ployed in  the  case  of  any  area  generated  by  a  straight  line 
whose  direction  is  invariable. 


130  GEOMETRICAL  APPLICATIONS.  [Art.  103. 

Let  AB  be  the  generating  line,  and  let  R  be  its  intersection 
with  a  fixed  line  CD,  to  which  it  is  always 
perpendicular.  Suppose  R  to  move  uni- 
formly along  CD,  and  let  RS  be  the  space 
described  by  R  in  the  interval  of  time,  dt. 
Then  the  value  of  the  differential  of  the 
area,  at  the  instant  when  the  generating  line 
passes  the  position  AB,  is  the  area  which 
would  be  generated  in  the  time  dt,  if  the 
rate  of  the  area  were  constant.  This  rate 
would  evidently  become  constant  if  the  generating  line  werev 
made  constant  in  length  ;  and  therefore  the  differential  is  the 
rectangle,  represented  in  the  figure,  whose  base  and  altitude 
are  AB  and  RS ;  that  is,  it  is  the  product  of  the  generating  line, 
and  the  differential  of  its  motion  in  a  direction  perpendicular  to 
its  length. 

104.  In  the  algebraic  expression  of  this  principle,  the  inde- 
pendent variable  is  the  distance  of  R  from  some  fixed  origin 
upon  CD,  and  the  length  of  AB  is  to  be  expressed  in  terms 
of  this  independent  variable. 

When  the  curve  or  curves  defining  the  length  of  AB  are 
given  in  rectangular  coordinates,  CD  is  generally  one  of  the 
axes;  thus,  if  the  generating  line  is  the  ordinate  of  a  curve, 
the  differential  is  y  dx,  as  shown  in  Art.  3.  It  is  often,  how- 
ever, convenient  to  regard  the  area  as  generated  by  some 
other  line. 

For  example,  given  the  curve  known  as  the  witch,  whose 
equation  is 

fx  —  2ay*  +  40?*  =  o (i) 

This  curve  passes  through  the  origin,  is  symmetrical  to  the 
axis  of  x,  and  has  the  line  x  =  2a  for  an  asymptote,  since 
x  =  2a  makes  y  =  ±  oo  . 

Let  the  area  between  the  curve  and  its  asymptote  be  re- 


§  VIII.]    AREAS  GENERATED  BY    VARIABLE  LINES.  131 

quired.     We  may  regard   this  area  as  generated  by  the  line 
PQ  parallel  to  the  axis  of  x,  y  being  taken 
as  the  independent  variable.     Now 

PQ  —  2a  —  xy 
hence  the  required  area  is 

A  =  \      (2a  -  x)  dy .     .     .     .     (2) 

J    —  00 

From  the  equation  (i)  of  the  curve,  we 
have 


x  = 


,  od  FIG.  10. 

whence        2a  —  x  —  —5— — — 2 , 


and  equation  (2)  becomes 

,dy    s^/^tan-1-^]      =4710?. 

o/+4tf  2tf_|_00 

Oblique  Coordinates. 

105.  When  the  coordinate  axes  are  oblique,  if  a  denotes 
the  angle  between  them,  and  the  ordinate  is  the  generating 
line,  the  differential  of  its  motion  in  a  direction  perpendicular 
to  its  length  is  evidently  sin  a-dx ;  therefore,  the  expression 
for  the  area  is 


A  =  sin  a  \  y  dx. 


132  GEOMETRICAL  APPLICATIONS.  [Art.  10$. 

As  an  illustration  let  the  area  between  a  parabola  and  a  chord 
passing  through  the  focus  be  required.  It  is  shown  in  treatises 
on  conic  sections,  that  the  value  of  a  focal  chord  is 


AB  =  4acosec?(x,      .      .     .      (i) 

x 

where  a  is  the  inclination  of  the  chord 

to  the  axis  of  the  curve,  and  a  is  the 
distance  from  the  focus  to  the  vertex. 
It  is  also  shown  that  the  equation  of 
the  curve  referred  to  the  diameter 
which  bisects  the  chord,  and  the  tan- 
gent at  its  extremity  which  is  parallel  to  the  chord  is 

y*  =  4a  cosec2  a-x (2) 

The  required  area  may  be  generated  by  the  double  ordi- 
-nate  in  this  equation ;  and  since  from  (i)  the  final  value  of 
y  is  ±  2a  cosec2  a,  equation  (2)  gives  for  the  final  value  of  x 

OR  —  a  cosec2  a. 
Hence  we  have 


fa  cosec'a 

A  =  2  sin  a  I  y  dx, 

or  by  equation  (2) 


/•ac 

=  4  Va  \ 
Jo 


A  =     Va  \  \/x  dx  - 


8tf2  COS6C3 


3 

Employment  of  an  Auxiliary   Variable. 
i06.  We  have  hitherto  assumed  that,  in  the  expression 


§  VI  1  1  .]     EMPLO  YMENT  OF  AN  A  UXILIAR  Y  VARIABLE.        1  3  3 

x  is  taken  as  the  independent  variable,  so  that  dx  may  be 
assumed  constant  ;  and  it  is  usual  to  take  the  limits  in  such  a 
manner  that  dx  is  positive.  The  resulting  value  of  A  will 
then  have  the  sign  of  y,  and  will  change  sign  if  y  changes 
sign. 

It  is  frequently  desirable,  however,  as  in  the  illustration 
given  below,  to  express  both  y  and  dx  in  terms  of  some  other 
variable.  When  this  is  done,  it  is  to  be  noticed  that  it  is  not 
necessary  that  dx  should  retain  the  same  sign  throughout  the 
entire  integral.  The  limits  may  often  be  so  taken  that  the  ex- 
tremityof  the  generating  ordinate  must  pass  completely  around 
a  closed  curve,  and  in  that  case  it  is  easily  seen  that  the  com- 
plete integral,  which  represents  the  algebraic  sum  of  the  areas 
generated  positively  and  negatively,  will  be  the  whole  area  of 
the  closed  curve. 

107.  As  an  illustration,  let  the  whole  area  of  the  closed 
curve 


represented  in  Fig.  12,  be  required.    If  in  this  equation  we  put 

(x\^ 

(a)    :=Sm^' 


we  shall  have 


whence  x  =  a  sin8  ^,          and         y  —  b  cos3  V*     •    •     0) 

Therefore  \ydx=  ^ab  cos4  fy  sin2  ip  cfy* 


134 


GEOMETRICAL  APPLICATIONS. 


[Art.  107. 


Now  if  in  this  integral  we  use  the  limits  o  and  2?r,  the  point 

determined  by  equation  (i)  de- 
scribes the  whole  curve  in  the 
direction  A  BCD  A.  Hence  we 
have  for  the  whole  area 


f27r 
A  —  ^ab     cos4  ip  sin8  ty  dif>, 

Jo 

and  by  formula  (0 


The  areas  in  this  case  are  all  generated  with  the  positive 
sign,  since  when  y  is  negative  dx  is  also  negative.  Had  the 
generating  point  moved  about  the  curve  in  the  opposite  direc- 
tion, the  result  would  have  been  negative. 


Area  generated  by  a  Rotating  Line  or  Radius  Vector. 

108.  The  radius  vector  of  a  curve  given  in  polar  coordinates  is 
a  variable  line  rotating  about  a  fixed  extremity.     The  angular 

rate  is  denoted  by  -r-   and   may 


dt 


be    re- 


garded as  constant,  but  then  the  rate  at 
which  area  is  generated  by  the  radius 
vector  OP,  Fig.  13,  will  not  be  constant, 
since  the  length  OP  is  not  constant. 
The  differential  of  this  area  is  the 
area  which  would  be  generated  in  the  0 
time  dt,  if  the  rate  of  the  area  were  con- 
stant ; 


FIG.  13. 
that  is  to  say,  if   the   radius  vector  were  of  constant 


VIII.]       AREAS  GENERATED  BY  ROTATING  LINES. 


'35 


length.     It  is  therefore  the  circular  sector  OPR  of  which  the 
radius  is  r  and  the  angle  at  the  centre  is  d6. 

Since  arc  PR  =  r  d0, 

sector  OPR  =  -rtd8', 


therefore  the  expression  for  the  generated  area  is 


0) 


109.  As  an  illustration,  let  us 
find  the  area  of  the  right-hand  loop 
of  the  lemniscate 

i^—cf1  cos  2,0. 


FIG.  14. 


The  limits  to  be  employed  are   those    values  of   6  which 

7t  .  7t 

make  r  —  o;  that  is and  -. 

4          4 

Hence  the  area  of  the  loop  is 


2  J_» 

4 


110.  When  the  radii  vectores,  r2  and  rv  corresponding  to  the 
same  value  of  6  in  two  curves,  have  the  same  sign,  the  area 
generated  by  their  difference  is  the  difference  of  the  polar  areas 
generated  by  r,  and  r2.  Hence  the  expression  for  this  area  is 


A  =  l-(r?  -  r?)  d8  .......  (2) 


136 


GEOME  TRICA  L   A  P PLICA  TIONS. 


[Art.  III. 


HI.  Let   us    apply  this  formula    to    find    the   whole    area 
between  the  cissoid 

'»  r-i  =  2a  (sec  6  —  cos  #), 

Fig.    15,  and  its  asymptote  £PZ,  whose 
polar  equation  is 

r2  =  2a  sec  6. 

—     One  half  of  the  required  area  is  generated 
by  the  line  P\P&  while  6  varies  from  o  to 

—  7i.     Hence  by  the  formula 
2 


A  = 


(2  -  cos20)  dd=~ 


FIG.  15. 


Therefore  the  whole  area  required  is 


Transformation  of  the  Polar  Formulae. 

112.  In  the  case  of  curves  given  in  rectangular  coordinates, 
it  is  sometimes  convenient  to  regard  an  area  as  generated  by  a 
radius  vector,  and  to  use  the  transformations  deduced  below 
in  place  of  the  polar  formulae. 


Put 


y  —  mx; .     (i) 


now  taking  the  origin  as  pole  and  the  initial  line  as  the  axis 
of  x,  we  have 

x  —  r  cos  0,  y  =  r  sin  6 :    .     «     .     (2) 


therefore 
and 


=     =  tan  0, 
x 


dm  =  sec2  8 


-      (3) 


§  VIII.]      TRANSFORMATION  OF  THE  POLAR  FORMULAS.      137 
From  equations  (2)  and  (3), 

x*  dm  =  r*  d&; 
therefore  equation  (i)  of  Art.  108  gives 


(4) 
In  like  manner,  equation  (2)  Art.  no  becomes 

4  ^U«  -*/)<*"  ......  (5) 

113.  As  an  illustration,  let  us  take  the  folium 

o  ......     (i) 


Putting  y  =  mx,  we  have 

Xs  (  I  +  nf]  —  ^amx*  =o  ......     (2) 


This  equation  gives  three  roots  or  values  of  x>  of  which  two 
are  always  equal  zero,  and  the  third  is 


x- 


whence  y—  — - „ ........ (5) 

i  +  nP 

These  are  therefore  the  coordinates  of  the  point  P  in  Fig.  16. 
Since  the  values  of  x  and  y  vanish  when  m  =  o,  and  when 
m  —  oo ,  the  curve  has  a  loop  in  the  first  quadrant.  To  find 


138 


GEOMETRICAL  APPLICATIONS. 


[Art.  113. 


the  area  of  this  loop  we  therefore  have,  by  equation  (4)  of  the 
preceding  article, 


_  9^  f°° 
2    J0i 


21+ 


114.  The  area  included  between  this  curve  and  its  asymr> 
p      tote  may  be  found  by  means  of  equation 
(5),  Art.  112.     The  equation  of  a  straight 
line  is  of  the  form 


y  =  mx  +  b, 


o\   o\ 
c 

FIG.  16. 


and  since  this  line  is  parallel  to  y  — 
the  value  of  m  for  the  asymptote  must  be 

that  which  makes  x  and  y  in  equations  (4)  and  (5)  infinite  ; 

that  is,  m  =  —  i  ;  hence  the  equation  of  the  asymptote  is 


y  +  x  =  b, 


(6) 


in  which  b  is  to  be  determined.  Since  when  m  =  —  i,  the 
point  P  of  the  curve  approaches  indefinitely  near  to  the  asymp- 
tote, equation  (6)  must  be  satisfied  by  P  when  m=  —  I. 
From  (4)  and  (5)  we  derive 


y  +  x  = 


m 


i  +  nr      i  —  m  +  m2 
whence,  putting  m=  —  i,  and  substituting  in  equation  (6) 

—  a  =  b, 
the  equation  of  the  asymptote  AB,  Fig.  16,  is 

y  +  ^  =  -  a  ........    (7) 


§  VIII.]     TRANSFORMATION  OF  THE  POLAR  FORMULAE.       139 

Now,  as  m  varies  from  —  oo  to  o,  the  difference  between  the 
radii  vectores  of  the  asymptote  and  curve  will  generate  the 
areas  OBC  and  ODA,  hence  the  sum  of  these  areas  is  repre 
sented  by 


A=-\°    (*$-*¥)  dm, 

2J-00 


in  which  xz  is  taken  from  the  equation  of  the  asymptote  (7), 
and  x^  from  that  of  the  curve. 
Putting  y  =  mx,  in  (7),  we  have 


i  +  m  ' 
and  the  value  of  xv  is  given  in  equation  (4).     Hence 


.  _  a*  , 

~  *""         a 


+  nf 


i    T 

i  +  m  J_0 


2  +  m  — 


2        i  +  n          _         2    i  —  m 


Adding  the  triangle  OCD,  whose  area  is  \cP,  we  have  for  the 
whole  area  required  ftf2. 


*  This  reduction  is  given  to  show  that  the  integral  is  not  infinite  for  the 
value  m  =  —  i,  which  is  between  the  limits.     See  Art.  82. 


140  GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 

Examples  VIII. 

i.  Find  the  area  included  between  the  curve 


and  the  axis  of  x.  — 

12 

2.  Find  the  whole  area  of  the  curve 

ay  =  *•  (a*  -  x*).  — 

4 

3.  Find  the  area  of  a  loop  of  the  curve 


4.  Find  the  area  between  the  axes  and  the  curve 


5.  Find  the  area  between  the  curve 


and  one  of  its  asymptotes.  20*. 

6.  Find  the  area  between  the  parabola  y*  =  ^ax  and  the  straight 

80s 
line  y  =  x.  —  . 

3 

y.  Find  the  area  of  the  ellipse  whose  equation  is 


§  VIII.]  EXAMPLES.  141 

8.  Find  the  area  of  the  loop  of  the  curve 

Of  =  (*-«)(*-*)', 

8  (b  -  a)\ 

tn  which  c  >  o  and  b  >  a.  —J-» 

i$Vc 

9.  Find  the  area  of  the  loop  of  the  curve 


1050* 

10.  Find  the  area'included  between  the  axes  and  the  curve 


20 


ii.  If  «  is  an  integer,  prove  that  the  area  included  between  the 
axes  and  the  curve 


is  *=^    :'    \  i**. 

zn  (2n  —  !)•••(«+  i) 


12.  If  n  is  an  odd  integer,  prove  that  the  area  included  between 
the  axes  and  the  curve 


/-,.\  • 

G)' 


_ 

la  ^1  —  7  •  . 

2«  (2«  —  2)  •  •  •  2       2 


142  GEOMETRICAL  APPLICATIONS.  [Ex.  VII L 

13.  In  the  case  of  the  curtate  cycloid 

x  =  aib  —  b  sin  ^,  y  —  a  —  b  cos  ipt 

find  the  area  between  the  axis  of  x  and  the  arc  below  this  axis. 

(20*  +  P)  cos-1 1  -  3a  V(?  -  A 

14.  If  b  =  %a7T,  show   that  the   area   of  the  loop  of  the  curtate 
cycloid  is 


-OH- 


15.  Find  the  area  of  the  segment  of  the  hyperbola 

x  —  a  sec  ip,  y  =  b  tan  ^, 

cut  off  by  the  double  ordinate  whose  length  is  2b. 

ab\   1/2  —  log  tan  ^—-    . 

1 6.  Find  the  whole  area  of  the  curve 

r*  =  a*  cosa  9  +  t?  sin'  9.  -  (a*  +  £'). 

2 

17.  Find  the  area  of  a  loop  of  the  curve 

>         »       o          ..    .  .  „                   ab       (a  —  b )        _,  a 
r*  =  a1  cos  0  —  b  sm*  9.  -  +  5 1  tan   '  - , 

22  O 

1 8.  Find  the  areas  of  the  large  and  of  each  of  the  small  loops  of 
the  curve 

r  =  a  cos  9  cos  29  ; 


§  VIII.]  EXAMPLES.  .  143 

and  show  that  the  sum  of  the  loops  may  be  expressed  by  a  single 
integral. 

no1   ,  a'  7ta*      a* 

—•?-  +  -  ,     and       ---  . 
10        6  32       12 

19.  In  the  case  of  the  spiral  of  Archimedes, 

r  =  aB, 

find  the  area  generated  by  the  radius  vector  of  the  first  whorl  and 
that  generated  by  the  difference  between  the  radii  vectores  of  the  nth 
and  (n  +  i)th  whorl. 

2      3 

—  —       and      Sna'Tr3. 
3 

20.  Find  the  area  of  a  loop  of  the  curve 

TTO* 
r  =  a  sin  39.  —  . 

12 

21.  Find  the  area  of  the  cardioid 

r  =  40  sin2  £9.  6ita*. 

22.  Find  the  area  of  the  loop  of  the  curve 

cos  29  a1  (4  —  TT) 

r  =  a  --  -.  -  . 

cos  0  .         2 

23.  In  the  case  of  the  hyperbolic  spiral, 


show  that  the  area  generated  by  the  radius  vector  is  proportional  to 
the  difference  between  its  initial  and  its  final  value. 


144  GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 

24.  Find  the  area  of  a  loop  of  the  curve 

ir.cf 

r  =  a  cos  n  Q.  -  — 

4* 

N 

25.  Find  the  area  of  a  loop  of  the  curve 


. 

COS1  6  '  2 

26.  Find  the  area  of  a  loop  of  the  curve 

r*  sin  6  =  a*  cos  26. 

Notice  that  r  />  raz/  and  finite  from  6  =  —  to  6  =  —  ,  #;?</  //fotf   —  — 

4  4  J  sin  9 

is  negative  in  this  interval.  a*     ^2  —  log  (i  +  V^)     , 

27.  Find  the  area  of  a  loop  of  the  curve 


Transform  to  polar  coordinates. 


28,  In  the  case  of  the  li 

r~==  za  cos©  +  b, 

find  the  whole  area  of  the  curve  when  b  >  za  and  show  that  the  same 
expression  gives  the  sum  of  the  loops  when  b  <  2a, 


V1I1.J  EXAMPLES. 


29.  Find  separately  the  areas  of  the  large  and  small  loops  of  the 
limafon  when  b  <  2a. 


If  a  =  cos-I  ( 

\  t 

large  loop  =  (20"  +  £a)  a  +  — 


small  loop  =  (2as  +  F)  (n  -  a)  - 


30.  Find  the  area  of  a  loop  of  the  curve 


31.  Find  the  area  of  the  loop  of  the  curve 
2  cos  26—1 


r  =  a 


cos  9 


32.  Show  that  the  sectorial  area  between  the  axis  of  x,  the  equi- 
lateral hyperbola 


and  the  radius  vector  making  the  angle  9  at  the  centre  is  represented 
by  the  formula 

.        i         i  +  tan  9 


and  hence  show  that 

-  g-SA 


and  y  = 


If  A  denotes  the  corresponding  area  in  the  case  of  the  circle 

**+/=  i, 
we  have 

x  =  cos  2 A,  and  y  =  sin  2^4. 


146  GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 

In  accordance  with  the  analogy  thus  presented,  the  values  of  x  and  y  given 
above  are  called  the  hyperbolic  cosine  and  the  hyperbolic  sine  of  2  A.      Thus 


_ 

=  cosh  (2  A),  -  =  smh  (2  A). 

2  ^ 


33.  Find  the  area  of  the  loop  of  the  curve 


—  o. 


35.  Find  the  area  between  the  curve 


38.  Trace  the  curve 


•  y 

x—  za  sm— . 
x* 


and  find  the  area  of  one  loop. 


. 

34.  Find  the  area  of  the  loop  of  the  curve 


2«  +  I     „ 

and  its  asymptote^  •  -  a  . 


36.  Find  the  area  of  the  loop  of  the  curve 

y3  +  ax*  —  axy  —  o. 

37.  Find  the  area  of  a  loop  of  the  curve 


x*  +  y*  =0*xy.  -- 


Six.] 


VOLUMES  OF  GEOMETRIC  SOLIDS. 


IX. 

Volumes  of  Geometric  Solids. 

115.  A  geometric   solid    whose  volume   is    required   is    fre- 
quently defined  in  such  a  way  that  the  area  of  the  plane  sec- 
tion parallel  to  a  fixed  plane  may  be  expressed  in  terms  of 
the  perpendicular  distance  of  the  section  from  the  fixed  plane. 
When  this  is  the  case,  the  solid  is  to  be  regarded  as  generated 
by  the  motion  of  the  plane  section,  and  its  differential,  when 
thus  considered,  is  readily  expressed. 

116.  For  example,  let  us  consider  the  solid  whose  surface  is 
formed  by  the   revolution  of  the  curve  APB,  Fig.  17,  about 
the  axis  OX.     The  plane  section  per- 
pendicular to  the  axis  OX  is  a  circle; 

and  if  APB  be  referred  to  rectangu- 
lar coordinates,  the  distance  of  the 
section  from  a  parallel  plane  passing 
through  the  origin  is  x,  while  the 
radius  of  the  circle  isj.  Supposing 
the  centre  of  the  section  to  move 
uniformly  along  the  axis,  the  rate  at 
which  the  volume  is  generated  is  not 
uniform,  but  its  differential  is  the  vol- 
ume which  would  be  generated  while  the  centre  is  describing 
the  distance  dx,  if  the  rate  were  made  constant.  This  differen- 
tial volume  is  therefore  the  cylinder  whose  altitude  is  dx,  and 
the  radius  of  whose  base  isjj>.  Hence,  if  V  denote  the  volume, 

dV '=  rcy*dx. 

117.  As  an  illustration,  let  it  be  required  to  find  the  volume 
of  the  paraboloid,  whose  height  is  h,  and  the  radius  of  whose 
base  is  b. 


148  GEOMETRICAL  APPLICATIONS.  [Art.  1  1  7. 

The  revolving  curve  is  in  this  case  a  parabola,  whose  equa- 
tion is  of  the  form 


and  since  y  =  b  when  x  =  h, 

a 

&  —  4flh,  whence  4^  =  -7  ; 

ft 

the  equation  of  the  parabola  is  therefore 


Hence  the  volume  required  is 

,.  Jff*  n&h 

dx  =  7r—\    xdx= 


. 
h  Jo  2 

118.  It  can  obviously  be  shown,  by  the  method  used  in 
Art.  1  1  6,  that  whatever  be  the  shape  of  the  section  parallel  to 
a  fixed  plane,  the  differential  of  the  volume  is  the  product  of  the 
area  of  the  generating  section  and  the  differential  of  its  motion 
perpendicular  to  its  plane. 

If  the  volume  is  completely  enclosed  by  a  surface  whose 
equation  is  given  in  the  rectangular  coordinates  x,  y,  z,  and  if 
we  denote  the  areas  of  the  sections  perpendicular  to  the  axes 
by  Ax,  Ay,  and  Az,  we  may  employ  either  of  the  formulas 

V  =  \AX  dx,  V=  \Ay  dy,  V  =  IAZ  dz. 

The  equation  of  the  section  perpendicular  to  the  axis  of  x 
is  determined  by  regarding  x  as  constant  in  the  equation  of 
the  surface,  and  its  area  Ax  is  of  course  a  function  of  x. 


§  IX.]  VOLUMES  OF  GEOMETRIC  SOLIDS.  149 

For  example,  the  equation  of  the  surface  of  an  ellipsoid  is 
#     /     £ 

__  L.  -L_     _L-    _    -     T 

a*  +  &      <?~ 
The  section  perpendicular  to  the  axis  of  x  is  the  ellipse 


whose  semi-axes  are  -   y(a2  —  x^)  and  -  V(a2  —  x*). 
a  a     v 


Since  the  area  of  an  ellipse  is  the  product  of  n  and  its  semi- 
axes, 


The  limits  for  x  are  ±a,  the  values  between  which  x  must  lie 
to  make  the  ellipse  possible.     Hence 


nbc  (a      o 
=  ~3-        (a 

«2    J-aV 


119-  The  area  A*  can  frequently  be  determined  by  the  con- 
ditions of  the  problem  without  finding  the  equation  of  the 
surface.  For  example,  let  it  be  required  to  find  the  volume  of 
the  solid  generated  by  so  moving  an  ellipse  with  constant 
major  axis,  that  its  center  shall  describe  the  major  axis  of  a 
fixed  ellipse,  to  whose  plane  it  is  perpendicular,  while  the  ex- 
tremities of  its  minor  axis  describe  the  fixed  ellipse.  Let  the 
equation  of  the  fixed  ellipse  be 


150 


GEOME  TRIG  A  L   A  P  PLICA  TIONS. 


[Art.  iig. 


and  let  c  be  the  major  semi-axis  of  the  moving  ellipse.  The 
minor  semi-axis  of  this  ellipse  is  y.  Since  the  area  of  an 
ellipse  is  equal  to  it  multiplied  by  the  product  of  its  semi-axes, 
we  have 


Therefore  J 

hence,  see  formula  (M ), 


— 
a      - 


•rPabc 


The  Solid  of  Revolution  regarded  as  Generated  by  a 
Cylindrical  Siirface. 

120.  A  solid  of  revolution  may  be  generated  in  another 

manner,  which  is  sometimes  more 
convenient  than  the  employment 
of  a  circular  section,  as  in  Art.  1 16. 
For  example,  let  the  cissoid  POR, 
Fig.  1 8,  whose  equation  is 


-*)=**, 


revolve  about   its   asymptote   AB. 
The  line  PR,  parallel  to  AB  and 
terminated  by  the  curve,  describes 
FIG.  is.  a  cylindrical  surface.      If   we    con- 

ceive the  radius  of  this  cylinder  to 

pass  from  the  value  OA  —  2a  to  zero,  the  cylindrical  surface 
will  evidently  generate  the  solid  of  revolution.      Now  every 


§  IX.J  DOUBLE  INTEGRATION.  151 

point  of  this  cylindrical  surface  moves  with  a  rate  equal  to 
that  of  the  radius;  therefore  the  differential  of  the  solid  is 
the  product  of  the  cylindrical  surface,  and  the  differential  of 
the  radius.  The  radius  and  altitude  in  this  case  are 

PC=2a-  x,  and  PR  =  2yt 

f2lt 

therefore  V  =  4?r      (lax  —  x^x  dx. 

Jo 

Putting  x  —  a  =  a  sin  6, 

4 

IT 

F=  47m8  [2  w(cos2  0  +  cos2  0  sin  0)  dO  =  2n*<£. 

2 

Examples  IX. 

i.  Find  the  volume  of  the  spheroid  produced  by  the  revolution  of 
the  ellipse, 


about  the  axis  of  x. 


2.  Find  the  volume  of  a  right  cone  whose  altitude  is  a,  and  the 
radius  of  whose  base  is  b.  nab" 


3.  Find  the  volume  of  the  solid  produced  by  the  revolution  about 
the  axis  of  x  of  the  area  between  this  axis,  the  cissoid 

y9  (za  —  x)  =  x3, 
and  the  ordinate  of  the  point  (a,  a).  &a*7r  (log  2  —  f ). 


152  GEOMETRICAL    APPLICATIONS.  [Ex.  IX. 

4.  Find  the  volume  generated  by  the  revolution  of  the  witch, 

y*x  —  20}? 
about  its  asymptote. 
See  Art.  104. 

5.  The  equilateral  hyperbola 


revolves  about  the  axis  of  x  \  show  that  the  volume  cut  off  by  a  plane 
cutting  the  axis  of  x  perpendicularly  at  a  distance  a  from  the  vertex 
is  equal  to  a  sphere  whose  radius  is  a. 

6.  An  anchor  ring  is  formed  by  the  revolution  of  a  circle  whose 
radius  is  b  about  a  straight  line  in  its  plane  at  a  distance  a  from  its 
centre  :  find  its  volume.  271*  aF. 

7.  Express  the  volume  of  a  segment  of  a  sphere  in  terms  of  the 
altitude  h  and  the  radii  a^  and  a?  of  the  bases. 

^  (A1  +  3*i2  +  3*'). 

8.  Find  the  volume  generated  by  the  revolution  of  the  cycloid, 

x  =  a  (ty  —  sin  ip),  y  =  a  (i  —  cos^), 

about  its  base.  5  ?r  V. 

9.  The  area  included  between  the  cycloid  and  tangents  at  the 
cusps  and  at  the  vertex  revolves  about  the  latter  ;  find  the  volume 
generated. 

*V. 

10.  Find  the  volume  generated  by  the  revolution  of  the  part  of  the 
curve 


which  is  on  the  left  of  the  origin,  about  the  axis  of  x. 

it 

2 


§  IX.]  EXAMPLES.  153 

11.  The  axes  of  two  equal  right  circular  cylinders,  whose  common 
radius  is  a,  intersect  at  the  angle  a  •  find  the  volume  common  to  the 
cylinders. 

The  section  parallel  to  the  axes  is  a  rhombus.  i6a3 

3  sin  a 

12.  Find  the  volume  generated  by  the  revolution  of  one  branch  of 
the  sinusoid, 


about  the  axis  of  x. 


y  =  b  sin  —  , 
a 


13.  Find  the  volume  enclosed  by  the  surface  generated  by  the  revo* 
lution  of  an  arc  of  a  parabola  about  a  chord,  whose  length  is  2cy  per- 
pendicular to  the  axis,  and  at  a  distance  b  from  the  vertex. 


14.  Find  the  volume  generated  by  the  revolution  of  the  tractrix, 
whose  differential  equation  is 

^..  + .     y 

T      '  ~    -J-       //a  »\  t 


about  the  axis  of  x. 

Express  ny*  dx  in  terms  of  y. 

3 

15.  Find  the  volume  generated  by  the  curve 

xy*  =  40"  (20  —  x) 
revolving  about  its  asymptote.  4^*^'. 

16.  Express  the  volume  of  a  frustum  of  a  cone  in  terms  of  its 
height  hy  and  the  radii  a1  and  a,  of  its  bases. 

nh  , 


154  GEOMETRICAL   APPLICATIONS.  [Ex.  IX. 

17.  Find  the  volume  of  a  barrel  whose  height  is  zh,  and  diameter 
2^,  the  longitudinal  section  through  the  centre  being  a  segment  of  an 
ellipse  whose  foci  are  in  the  ends  of  the  barrel. 


18.  Find  the  volume  generated  by  the  superior  and  by  the  inferior 
branch  of  the  conchoid  each  revolving  about  the  directrix  ;  the 
equation,  when  the  axis  of  y  is  the  directrix,  being 

*y  =  (a  +  xy(p  -  **). 


19.  The  area  included  between  a  quadrant  of  the  ellipse 
x  =  a  cos  0,  y  =  b  sin  0, 

and  the  tangents  at  its  extremities  revolves  about  the  tangent  at  the 
extremity  of  the  minor  axis  ;  find  the  volume  generated. 


20.  An  ellipse  revolves  about  the  tangent  at  the  extremity  of  its 
major  axis  ;  express  the  entire  volume  in  the  form  of  an  integral, 
whose  limits  are  o  and  27T,  and  find  its  value.  2Tt*c?b. 

21.  Find  the  volume  generated   by  the  revolution  of  a  circular 
arc  whose  radius  is  a  about  its  chord  whose  length  is  zc. 

27ZV(l02  —  f)  j      //     a  a\      •        1   c 

—  ^—      —  -  —  2Tta   d(a   —  c]  sin"1  -. 
3  a 

22.  A  straight  line  of  fixed  length  zc  moves  with  its  extremities 
in  two  fixed  perpendicular  straight  lines  not  in  the  same  plane,  and 
at  a  distance  zb.     Prove  that  every  point  in  the  moving  line  de- 
scribes an  ellipse  in  a  plane  parallel  to  both  the  fixed  lines,  and  find 
the  volume  enclosed  by  the  generated  surface.  471  (f  —  tf)b 


§  X.]  DOUBLE  INTEGRALS.  155 

X. 

Double  Integrals. 

121.  Let  us  consider  the  expression 

,j2 

<t>(x,y)dy,      .......     (i) 

J  y\ 

in  which  the  limits  j/,  and  j2  may  be  any  functions  of  x.  In 
the  integration,  ^  is  to  be  treated  like  any  other  quantity 
independent  of  y  which  may  be  involved  in  the  expression 
(f>(x,  y]  ;  in  other  words,  as  a  constant  with  respect  to  the 
variable  y.  Thus  the  indefinite  integral  will  contain  both  x 
and  y  ;  but  since  the  definite  integral  is  a  function  not  of  the 
independent  variable  but  of  the  limits,  the  expression  (i)  is 
independent  of  y,  but  is  generally  a  function  of  x.  We  may 
therefore  denote  it  by  f(x),  and  write  it  in  place  of  f(x)  in  the 
expression  of  an  integral  in  which  x  is  the  independent  vari- 
able. Thus,  putting 

=/(*)>      ......      (2) 


i  y\ 


\f(x}dx=\    ^<p(x, 

•fa  *  **  y\ 


(3) 


In  the  last  expression,  -which  is  called  a  double  integral,  x 
andjj/  are  two  independent  variables.  It  is  to  be  noticed  that 
the  limits  of  the  ^-integration,  which  is  to  be  performed  first, 
may  be  functions  of  the  other  variable  x,  but  the  limits  of  the 
final  integration  must  be  constants,  that  is,  independent  both 
of  x  and  of  y. 

122.  In  accordance  with  Art.  99,  the  expression  (i)  is  the 

limit  of  -2  "0(.r,  y)  Ay,  when  Ay  is  diminished  without   limit, 
y\ 

it  being  assumed  that  (}>(x,  y)  is  finite  and  continuous  while  y 
passes  from  7,  to  /„.     Further,  assuming  this  to  be  the  case 


i56 


GEOMETRICAL    APPLICATIONS.  [Art.    122. 


for  all  values  of  x  ,from  a  to  b  (so  that/"(.r)  in  equation  (3)  is 

finite  and  continuous  for  all  values  concerned  in  the  .r-integra- 

g 
tion),  the  double  integral  is  the  limit  of  2af(x)  Ax,  when  Ax 

is  diminished  without  limit.      It  readily  follows  that  the  double 
integral  is  the  limit  of 


where  both  Ay  and  Ax  are  diminished  without  limit.* 

The  typical  term  fy(x,  y]  Ay  Ax  is  called  the  element  of  the 
sum,  and  in  like  manner  <f>(x,  y]  dy  dx  is  the  element  of  the  double 
integral.  As  mentioned  in  Art.  100,  in  forming  the  expres- 
sion for  the  element,  no  distinction  need  be  drawn  between  any 
values  of  x  and  y,  between  x  and  x  +  Ax,  y  and  y  -f  Ay,  re- 
spectively, because  these  distinctions  disappear  at  the  limit. 

Limits  of  the  Double  Integral. 

123.  In  discussing  the  limits  of  the  double  integral  (3), 
Art.  121,  it  is  convenient  to  consider 
first  the  simpler  expression 


dy  dx. 


FIG.  19. 


Performing  the  jj/-integration,   this    re- 
duces to  the  simple  integral 


(2) 


*  Denoting  the  difference  between  _/{*•)  and 


.  y)Ay,  of  which  ifis  ihe 


limit,  by  e,  e  is  a  quantity  which  vanishes  with  Ay.  Then  the  difference  be- 
tween ~2af(x)  Ax  and  2*2f^*#(.t,  y)  Ay  Ax  is  'S^Ax,  a  quantity  which  vanishes 

with  Ay  if  a  and  b  are  finite.  Hence,  in  this  case,  the  double  integral  which  is 
the  limit  of  the  first  of  these  sums  is  also  the  limit  of  the  second.  But  the  con- 
clusion does  not  follow  when  the  limits  are  infinite  ;  in  fact,  the  double  integral 
is  not  then  always  independent  of  the  mode  in  which  the  limits  become  infinite. 


§  X.]  LIMITS   OF   THE  DOUBLE   INTEGRAL.  157 

In  Fig.  19,  using  rectangular  coordinates,  let  OA  =  a, 
OB  —  b,  and  let  CD,  EF  be  the  curves  y  =  j,  ,  y  =  ja  ;  then 
(2)  is,  by  Art.  103,  the  expression  for  the  area  CDFE.  There- 
fore the  double  integral  (i)  is  represented  by  the  area  enclosed 
by  the  curves  y  =•  yl  ,  y  =  y^  and  the  straight  lines  x  =  a,  x  =  b. 

I24-.  In  order  that  the  double  integral  may  represent  the 
area  enclosed  by  a  single  curve  (like  the  dotted  line  of  Fig. 
19)  of  which  the  equation  is  known,  the  limits  jj/,  and  y^  must 
be  two  values  of  y  corresponding  to  ,the  same  value  of  x,  and 
the  limits  for  x  must  be  those  values  for  which  yl  and  y^  are 
equal.  Between  the  limiting  values  of  x  the  values  of  y  are 
real,  and  beyond  them  the  values  of  y  become  imaginary.  For 
example,  suppose  the  curve  to  be  the  ellipse 


2X  —  2.xy  +  y  —  4x  — 
solving  for  y, 

y  =  x+l±  ^(-^  +  6x  -  5)  =  x 
whence 


This  expression  is  real  for  all  values  of  x  between  I  and  5  ; 
hence  the  entire  area  is 


-  i)(5  -  *)]  dx*  =  4*. 


It  is  evident  that  we  might  equally  well  have  used  the  ex- 
pression 


f  q  fx*  (q 

A  =          dxdy  =  \(x,  —  x,)dy, 

itlxi  JP 


*  It  is  useful  to  notice  that  a  definite  integral  of  this  form,  by  a  familiar 
property  of  the  circle,  represents  the  area  of  the  semicircle  whose  diameter  is 
the  difference  between  the  limits. 


158  GEOMETRICAL   APPLICATIONS.  [Art.    124. 

in  which  x^  and  x^  are  obtained  as   functions  of  y,  by  solving 
the  equation  of  the  curve  for  x,  thus 

x  =  b(y  +  2)  ±  i  V(~  }'   +  8j  -  8)  ; 


and  the  limiting  values  of  y  are  those  obtained  by  putting  the 
radical  equal  to  zero,  namely,  p  =  4  —  2  4/2,  <?  =  4  +  2  |/2. 
Accordingly  the  same  area  is  expressed  by 


=/: 


-8), 


whence  y2  =  4?r  as  before. 

125.  It  appears  therefore  that  no  distinction  is  to  be  drawn 
between  the  expressions 


\dydx     and      I  \dxdy. 


We  may  in  fact  regard  either  one  as  representing  any  area 
whatever,  the  value  becoming  definite  only  when  we  assign  a 

defined    closed    contour   or   boundary   line;    just  as    \dx  may 

represent  a  line  of  any  length  measured  along  the  axis  of  x, 
and  is  only  definite  when  we  assign  the  limits  which  determine 
its  two  extremities.  Thus  the  contour  bears  the  same  relation 
to  the  double  integral  that  the  pair  of  limits  does  to  the  simple 
integral. 

When  the  boundary  of  a  given  area  is  made  up  of  lines 
having  different  equations,  it  is  not  generally  possible  to  ex- 
press the  area  by  means  of  a  single  double  integral.  This  was 
possible,  it  is  true,  in  the  case  of  the  area  enclosed  by  the  full 
line  in  Fig.  19,  because  the  equations  of  two  of  the  bounding 
lines,  x  =  a  and  x  =  b,  contained  only  one  variable,  and  the 
integration  with  respect  to  this  was  made  the  final  one  in  ex- 
pression (i).  But,  if  integration  with  respect  to  x  had  been 
performed  first,  it  would  have  been  necessary  to  break  the  area 


X.] 


THE   AREA    OF   INTEGRATION. 


159 


up  into  several  parts,  of  which  it  is  the  algebraic  sum.  In 
the  expressions  for  these  parts,  the  limits  for  x  would  be  taken 
from  the  equations  of  the  different  lines,  and  the  final  limits 
would  be  the  ordinates,  which  it  would  be  necessary  to  find 
(instead  of  the  known  abscissas),  of  the  intersections  C,  D,  E, 
and  F. 

126.   Returning  now  to  the  general  double  integral 

Oy* 
$(x,y)dydx, (i) 
y\ 

and    employing   rectangular   coordinates,  let  the  area  ASBR 
in  the  plane  of  xy,  Fig.  20,  be  that 
which    is    defined  by  the   limits  of 
the  given  integral,  in  other  words, 
the  area  represented  by 


dy  dx. 


(2) 


In   evaluating   the   double    integral 
(i)  with  the  given  limits,  the  Integra-      ' 
tion  of  the  element  <f)(x,y}dydx  is 
said  to  extend  over  the  area  ASBR 
represented  by  expression  (2). 

Now  let  us  construct  the  surface  whose  equation  is 


z  — 


,  y). 


(3) 


The  ^-coordinates  of  all  points  whose  projections  on  the  plane 
of  xy  are  on  the  curve  ASBR  lie  in  a  cylindrical  surface,  which 
in  Fig.  20  is  supposed  to  cut  the  surface  (3)  in  the  curve 
CQDP.  It  is  assumed  that  4>(x,y\  or  3,  remains  finite  and  con- 
tinuous while  the  independent  variables  x  and  y  vary  in  any 
way,  provided  the  point  (x,  y)  remains  within  the  area  ASBR. 
This  assumption  is  clearly  identical  with  that  made  in  Art.  122. 
A  definite  solid  will  then  be  enclosed  between  the  surface  (3), 


l6o  GEOMETRICAL   APPLICATIONS.  [Art.   126. 

the  plane  of  xy,  and  the  cylindrical  surface.     We  have  now  to 
show  that  this  solid  will  represent  the  double  integral  (i). 

Let  SRPQ  be  a  section  of  this  solid  by  a  plane  parallel  to 
that  of  yz,  so  that  in  it  x  has  a  constant  value.  The  ordinates 
of  the  points  R  and  S  are  the  values  of  y.t  and  y^  correspond- 
ing to  that  particular  value  of  x.  The  section  SRPQ  may  be 
regarded  as  generated  by  the  line  z,  while  y  varies  from  yl  to 
yt\  hence,  denoting  its  area  by  Ax,  as  in  Art.  118, 

f*1 

Ax=\z  dy. 

•  y\ 

Therefore  the  volume  is 

fb  fbtyt 

V=      Axdx  =          zdydx, 

la  J  ai  J»i 

which  is  identical  with  expression  (i). 

Change  of  the  Order  of  Integration. 

127.  It  is  obvious  that  the  order  of  integration  may  be 
reversed  as  in  Art.  124,  provided  that  the  integration  extends 
over  the  same  area.  Considering  the  corresponding  process  of 
double  summation,  we  may  be  said,  in  the  first  case,  to  sum  up 
those  of  the  elements  z  dy  dx  which  have  a  common  value  of  x 
into  the  sum  Axdx,  which  then  constitutes  the  element  of  the 
second  summation ;  while  in  the  second  case  we  first  sum  up 
those  of  the  original  elements  which  have  a  common  value  of 
y  into  the  element  Aydy,  which  is  afterward  summed  for  all 
values  of  y  within  the  given  limits. 

Since,  in  expression  (i),  Art.  126,  dx  is  a  constant  with 
respect  to  the  /-integration,  it  may  be  removed  to  the  left  of 
the  corresponding  integral  sign.  In  the  resulting  expression, 


f4      V* 
dx\ 

la        J  yi 


§x.j 


CHANGE    OF  ORDER  IN  INTEGRATION. 


161 


it  is  to  be  understood  that  because  the  /-integral  which  is  a 
function  of  x  follows  the  sign  of  ^-integration  it  is  "  under  " 
it,  or  subject  to  it.  This  notation  has  the  advantage  of  mak- 
ing it  perfectly  clear  to  which  variable  each  integral  sign  cor- 
responds.* 

128.  The  limits  of  a  double  integral  are  sometimes  given 
in  the  form  of  a  restriction  upon  the  values  which  the  inde- 
pendent variables  x  and  y  can  simultaneously  assume.     Sup- 
pose, for  example,  that  in  the  integration  neither  variable  can 
become    negative    or   exceed  a,  and 

that  y  cannot  exceed  x.     If  x  and/ 

are  coordinates  of  a  point  in  Fig.  21, 

the  restriction  is  equivalent  to  saying 

that  the  point  cannot  be  below  the 

axis  of  x,  to  the  right  of  the  line  x  =  a 

or  above  the  line  y  =  x.     Therefore 

the  area  of  integration  is  the  triangle 

OAB.    This  may  be  expressed  either 

by  giving  to  y  the  limits  zero  and  x, 

and  then  giving  to  x  the  limits  zero 

and  a ;  or,  reversing  the  order  of  integration,  by  giving  to  x  the 

limits  y  and  a,  and  then  giving  to  y  the  limits  zero   and  a. 

Thus  the  area  of  integration  can  be  represented  by  either  of 

the  expressions 

dydx>     or  dx  dy. 

•  oJo  1  o»  y 

129.  As  an  illustration,  let  us  take  the  integral 

U=  I   I  sin'1/  |/(i  —  x)  \/(x  —y]dydx. 

•  oJ  o 


FIG.  21. 


*  We  have  in  the  preceding  articles  supposed  the  first  written  integral  sign 
to  correspond  to  the  last  written  differential  when  the  pair  of  differentials  is 
written  last ;  but  writers  are  not  uniform  in  this  respect. 


l62  GEOMETRICAL  APPLICATIONS.  [Art.  1  29. 

The  /-integration,  which  is  here  indicated  as  the  first  to  be 
performed,  namely,  the  integration  of  \/(x  —  y)  sin"1  y  dy  is 
impracticable.  Noticing,  however,  that  the  ^-integration  in 
the  given  expression  is  of  a  known  or  integrable  form,  we  change 
the  order  of  integration,  determining  the  new  limits  so  as  to 
represent  the  same  area  of  integration.  Since  this  area  is  that 
indicated  in  Fig.  21,  when  a  =  i,  we  thus  obtain 

(7=     sirr1/^!    ^/[(i  —  x](x  —  y}\dx. 

Jo  J  y 

The  value  of  the  ^-integral  is  ^TT(I  —  /)2;  hence 


U  =  ~^\  sin-'j/(i  — 

°J  0 

Finally,  integrating  by  parts  and  then  putting  y  =  sin  6, 


-sn 


24o4/(i-/)       240  24_4 

130.  In  a  case  where  the  first  integration  can  be  effected 
with  respect  to  either  variable,  it  may  happen  that  in  one  case 
only  are  the  limits  such  as  to  make  the  second  integration 
possible.  Given,  for  example, 

f00  f3 
U—\        e-xydydx,  ......     (i) 

J  o  J  a 

in  which  a  and/?  are  both  positive.     Integration  for/  gives 


u= 


and  for  this  form  the  second  integration  is  impracticable.     But, 
integrating  expression  (i)  first  with  respect  to  x,  we  have, 


§  X.]  TRIPLE  INTEGRALS.  163 

owing  to  the  special  values  (zero  and   infinity)   of  the  limits, 
the  simple  form 


The  double  integration  in  this  example  extends  over  the 
infinite  strip  of  area  between  the  parallel  lines  y  =  /3,  y  =  a  and 
on  the  right  of  the  axis  of  y,  yet  we  have  obtained  a  finite 
result.  It  is  plain  that  this  would  not  have  been  possible 
but  for  the  fact  that  the  element  e~xy  dy  dx  approaches  to  the 
limit  zero,  when  x  increases  without  limit  (y  remaining  between 
finite  limits) ;  that  is,  the  element  vanishes  for  all  the  infinitely 
distant  points  which  are  included  in  the  area  of  integration. 

The  fact  that  we  have  thus  obtained  the  value  of  the 
definite  simple  integral  (2),  although  the  corresponding  in- 
definite integral  could  not  be  found,  will  be  noticed  in  con- 
nection with  the  methods  of  evaluating  definite  integrals. 

Triple  Integrals. 
131.  An  expression  of  the  general  form 

Dy-tf't 
<}>(x ,  y ,  z)  dz  dy  dx (i) 
y\i*i 

is  called  a  triple  integral.  In  the  first  integration  the  limits 
#,  and  z^  are  in  general  functions  of  x  and  y.  In  the  next,  the 
limits  j,  and^a  are  in  general  functions  of  x;  and  in  the  final 
integration  the  limits  are  constants. 

It  is  readily  seen  that  the  triple  integral,  like  the  double  in- 
tegral considered  in  Art.  122,  is  the  limit  (where  Ax,  Ay,  and  Az 
diminish  without  limit)  of  the  result  of  a  corresponding  sum- 
mation. Accordingly  (f>  (x,y,  z)  dz  dy  dx  is  called  the  element 
of  the  integral  (i). 


164  GEOMETRICAL   APPLICATIONS.  [Art.   131. 

Consider  now  the  triple  integral  with  the  same  limits,  but 
having  the  simpler  element  dz  dy  dx ;  the  ^-integration  can 
here  be  effected  at  once,  and  we  have 

f  *  pa 

OS  dydx=\        (z^—  #,)  dy  dx.  ...     (2) 

JaJjX, 

Now  supposing  x,  y,  and  z  to  be  rectangular  coordinates  in 
space,  the  last  expression  represents  the  difference  between 
two  solids  defined  as  in  Art.  126;  that  is,  the  triple  integral  in 
equation  (2)  represents  the  volume  included  between  the  surfaces 
z  —  #,,  z  =  z,,  and  the  cylindrical  surface  wJwse  section  in  the 
plane  of  xy  is  the  contour  determined  by  the  limits  for  y  and  x. 

132.  When  the  volume  is  completely  enclosed  by  a  surface 
defined  by  a  single  equation  or  relation  between  x,y,  and  z,  the 
case  is  analogous  to  that  of  the  area  discussed  in  Art.  124;  sl 
and  zt  will  be  two  values  of  z  corresponding  to  the  same  values 
of  x  and  y,  and  the  limits  for  x  and  y  must  be  determined  by 
the  area  within  which  the  value  of  z^  —  z^  is  real.  This  area 
is  the  section  with  the  plane  of  xy  of  a  cylinder  which  circum- 
scribes the  given  volume. 

In  this  case  it,  is  plain  that  we  may  perform  the  integrations 
in  any  order,  provided  we  properly  determine  the  limits. 
Considering  the  corresponding  process  of  summation,  we  may 
be  said  in  equation  (2)  to  have  summed  those  of  the  ultimate 
elements  which  have  common  values  of  x  and  y  into  the  linear 
element  (z^  —  £,)  dy  dx.  If  the  integration  for  y  is  effected 
next,  we  combine  such  of  these  last  elements  as  have  a  com- 
mon value  of  x  into  the  areal  element  Axdx  which  is  often 
called  a  lamina.  Thus  each  of  the  formulae  in  Art.  118  is  the 
result  of  performing  two  of  the  integrations  in  the  general 
expression  for  a  volume,  namely, 

V=  [\\dxdydz. 


§  X.]  THE    VOLUME   OF  INTEGRATION.  165 

In  any  case  where  the  result  of  two  integrations  (that  is,  the 
area  of  a  section)  is  known,  we  may  of  course  take  advantage 
of  this  and  pass  at  once  to  the  simple  integral,  as  in  Art.  118, 
where  in  finding  the  volume  of  the  ellipsoid  we  regarded  the 
expression  for  the  area  of  an  ellipse  as  known. 

Integration  over  a  Known  Volume. 

133.  In  the  general  expression,  (i)  of  Art.  131,  the  integra- 
tion is  said  to  extend  over  the  volume  defined  by  the  limits,  that 
is,  the  volume  represented  by  either  member  of  equation  (2). 
It  is  sometimes  possible,  in  the  case  of  a  triple  integral  of 
the  more  general  form,  to  take  advantage  of  our  knowledge  of 
the  geometrical  solid  over  which  the  integration  extends.  For 
example,  let  it  be  required  to  evaluate  the  integral 


u  = 

for  all  values  of  x,  y,  and  z  such  that 


does  not  exceed  unity,  that  is  to  say,  when  the  integration  ex- 
tends over  the  volume  of  the  ellipsoid  considered  in  Art.  118. 
We  here  perform  the  integration  for  y  and  z  first,  because  we 
see  that  the  result  will  be  the  section  Ax  of  this  solid.  This 
reduces  the  integral  to  the  simple  form 

TT  J  Ttbc  ta      „,     2  y.. 

I   I      —  ..        I     A-*     /I          SJ  T   I  •V"-/    »*      ..  r        -V*  lyY'** 

(_/    -         |  X,   yi  y.  U,*L  - — —    I     X,    \Cl     —   X    \U,X 

a?  }-a 

=  nctbcC2-  --}  = 


166  GEOMETRICAL   APPLICATIONS.  [Art.   134, 

134- .  A  transformation  of  variables  will  sometimes  enable  us 
to  make  use  of  geometrical  considerations.  For  example,  let 
us  find  the  value  of 

x 

dxdy  dz 
\/(xyz) 

for  all  positive  values  of  the  variables  whose  sum  does  not  ex- 
ceed a ;  or,  as  it  may  be  written, 

(adx  r~*  dy  ta-*-y  dz 

(^/    . 


t 

~  J 


4/jo         V2 

If  we  put  V  '  x  =  £,   \/y  =  rf,   ^z  =  C,  we  have 
U  =  sf   f  f  d£dnd£, 

J   O  •  O  J  o 

and  the  condition  determining  the  upper  limits  is 

&  +  jf  +  ?  =  a. 

The  last  integral  (regarding  &,,  rf,  and  C  as  a  new  set  of  rec- 
tangular coordinates)  represents  one  octant  of  a  sphere  whose 
radius  is  4/0  ;  hence 

U  =  $na*. 

Representation  of  a  Triple  Integral  by  a  Mass  of 
Variable  Density. 

135.  We  may,  in  any  triple  integral  of  the  general  form 

J  0(>,  y,  z)  dx  dy  dz, 
suppose  the  function  <p(x,  y,  z]  to  represent  the  density  of  the 


§  X.]      TRIPLE  INTEGRAL  REPRESENTED  BY  A   MASS.       l6/ 

geometric  element  dx  dy  dz  at  the  point  (x,  y,  z)*  so  that  the 
element  of  the  integral  is  the  element  of  a  mass  of  variable 
density  occupying  the  space  defined  by  the  limits  of  the  inte- 
gral. This  mass  will  therefore  represent  the  value  of  the 
triple  integral. 

For  example,  the  integral  U  of  Art.  134  extends  over  the 
volume  of  integration  enclosed  by  the  three  coordinate  planes 
and  the  plane 

x  +  y  +  z  =  a; 

and,  since  <p(x,  y,  z)  =  (xyz)~*,  the  indefinite  integral  repre- 
sents a  mass  whose  density  varies  inversely  as  the  square  root 
of  the  product  of  the  coordinates,  and  is  unity  at  the  point 
(i,  I,  i).  Therefore  the  definite  integral  U  represents  the  mass 
of  the  tetrahedron  or  triangular  pyramid  bounded  as  above  and 
having  this  law  of  variable  density. 

It  may  be  noticed  that,  in  the  transformation  employed  in 
evaluating  U,  the  solid  was  converted  without  change  of  mass 
into  an  octant  of  a  sphere  having  the  uniform  density  8. 

136.  When  the  variable  corresponding  to  the  integration 
indicated  as  the  final  one  is  not  involved  in  the  limits  of  either 
of  the  other  variables,  none  of  the  limits  will  be  changed  if  we 
effect  the  integration  with  respect  to  this  variable  first.  For 
example,  given  the  integral 

f*      ra      r  v(<*2  -  y*) 
dx\dy\  (x2  +  /  +  &)dz.     .     .     (i) 

o   '     Jo        J-VCa'-jr1) 

Because  x  does  not  occur  in  the  limits  for  y  and  £,  the  yz- 
integration  extends  over  the  same  area,  no  matter  what  the 

*  In  the  element  of  the  sum  of  which  the  triple  integral  is  the  limit,  (f>(x,y,  z) 
is  to  be  taken  as  the  average  density  of  the  element  Ax  Ay  Az,  so  that 
<p(x,  y,  z)Ax  Ay  Az  shall  be  the  mass  of  the  geometric  element,  that  is  to  say, 
the  element  of  mass.  But  this  average  density  is  the  density  at  some  point 
•within  the  element  Ax  Ay  Az,  and  the  distinction  between  all  such  points  dis- 
appears at  the  limit  as  in  the  case  of  the  simple  integral.  See  Art.  100. 


l£8  GEOME7^JCAL   APPLICATIONS.  [Art.   136. 

value  of  x  ;  namely,  in  this  case,  a  semicircle  whose  radius  is 
a.  This  is  as  much  as  to  say  that  the  volume  of  integration  is 
bounded  by  a  cylindrical  surface,  whose  elements  are  parallel 
to  the  axis  of  x,  together  with  planes  parallel  to  the  plane  of 
yz*  (In  the  example,  it  is  the  half  of  a  circular  cylinder  of 
height  //.)  Hence  we  may  write 


fa         t  tf(a*  -  y*)  f 

U=  I  dy\  dz\ 

Jo       J-VW-y*)      Jo 

h*  e  "  f  V(«»  -  r1)  faf  V(a*-y*) 

=  -  dzdy  +  h\  (f  +  ^dzdy.      (2) 

3  JoJ  -V(a*-j,*)  J  0  J  _  V(«»  -  >2) 


The  first  of  these  double  integrals  is  the  area  of  the  semi- 
circle mentioned  above     Hence 


2/1 


fa  "21  fa 

1  /  !/(«2  -  f}dy  +-    \(d*  - 

Jo  3   J  o 


6 
and  finally,  U  =  -^Tt/ia^a2  -f- 


Examples  X. 

1.  Find  the  volume   cut  from  a  right   circular  cylinder  whose 
radius  is  a,  by  a  plane  passing  through  the  centre  of  the  base  and 

making  the  angle  a  with  the  plane  of  the  base.  2   , 

-a   tan  a, 

3 

2.  Show  that  the  integral  of  x  dx  dy  over  any  area  symmetrical 
to  the  axis  of  y  vanishes.     Interpret  the  result  geometrically,  and 
apply  to  find  the  integral  of  (c  +mx  +  ny)dx  dy  over  the  ellipse 

«f/  +  tfx*  =  (?b\  nabc. 

*  The  case  is  analogous  to  that  of  the  double  integral  when  both  pairs  of 
limits  are  constants,  that  is,  when  the  area  of  integration  is  a  rectangle. 


§  X.]  EXAMPLES.  169 

3.  Show  that  the  volume  between  the  surface 

zn  =  a  v  +  jy 

and  any  plane  parallel  to  the  plane  of  xy  is  equal  to  the  circumscrib- 
ing cylinder  divided  by  n  +  i. 

4.  Find  the  volume  enclosed  by  the  surface  whose  equation  is 

x*       ya       2* 

-»  +  "71  +  -4  =  I- 


»  4         -  —  . 

a        b        c  5 

5.  A  moving  straight  line,  which  is  always  perpendicular  to  a 
fixed  straight  line  through  which  it  passes,  passes  also  through  the 
circumference  of  a  circle  whose  radius  is  a,  in  a  plane  parallel  to  the 
fixed  straight  line  and  at  a  distance  b  from  it;  find  the  volume  en- 
closed by  the  surface  generated  and  the  circle.  ncfb 

2 

6.  A  cylinder  cuts  the  plane  of  xy  in  an  ellipse  whose  semi-axes 
are  a  and  b,  and  the  plane  of  xz  in  an  ellipse  whose  semi-axes  are  a 
and  c,  the  elements  of  the  cylinder  being  parallel  to  the  plane  of  yz  ; 
find  the  volume  of  the  portion  bounded  by  the  semi-ellipses  and  the 

surface.  2 

-aoc. 
3 

7.  Find  the  volume  enclosed  by  the  surface 

£"...!?-£ 

J1**"* 

Ttabc 
and  the  plane  x  =  a.  ---  . 

2 

8.  Find  the  volume  enclosed  by  the  surface 


Find  Az  as  in  Art.  107. 

35 

9.  Find  the  volume  between  the  coordinate  planes  and  the  surface 


go 


GEOMETRICAL   APPLICATIONS.  [Ex.  X. 

10.  Find  the  volume  cut  from  the  paraboloid  of  revolution 

y  4   z*  =  ^ax 
by  the  right  circular  cylinder 

x*  +  y  =  2ax, 

whose  axis  intersects  the  axis  of  the  paraboloid  perpendicularly  at 
the  focus,  and  whose  surface  passes  through  the  vertex. 

3    ,    i6a' 

-\  --  . 


ii.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  a 
right  circular  cylinder  whose  radius  is  b,  and  whose  axis  passes  through 
the  centre  of  the  sphere. 


»  _  /  j  _  m 

12.  Prove  that  the  volume  generated  by  the  revolution  about  the 
axis  of  y  of  the  area  between  the  equilateral  hyperbola 


and  the  double  ordinate  2yl  is  equal  to  the  sphere  of  radius  yt.    [Ex. 
1  1  shows  that  the  circle  has  the  same  property.] 

13.  Change  the  order  of  integration  in  the  double  integral 

n-za  —  y 
(f)(x,y)dx(ty. 
y 

t  a  t  x  tia  tia  —  x 

(p(x,y)dy  dx  +  </>(x,  y]dy  dx. 

JoJ°  JrtJo 

14.  Evaluate  the  integral 

taf*a-y         dx  dy 

\o\y        V(x*-xy}'  2*/2l°S(      ^2}a' 

15.  Integrate  xydydx  over  the  area  of  the  circle 

(x  -  /O3  +  (y  -  k)*  =  a\  ncthk. 


§  X.]  EXAMPLES. 


16.  Show  that  the  integral  of  the  element  in  Ex.  15  over  any 
square  circumscribing  the  given  circle  is  \a^hk. 

17.  Integrate  x*y  dy  dx  over  the  circle  of  Ex.  15,  and  over  the 
circumscribing  square  with  sides  parallel  to  the  axes. 

na*hk(h*  +  fa1);    ^hk(tf  +  c?}. 

1  8.  Evaluate      i  \xyzdxdy  dz  for  all  positive  values  of  x,y,  and 

z  whose  sum  is  less  than  a;  also  for  all  positive  values  the   sum  of 
whose  squares  is  less  than'  0s.  a"       a" 

720'    48' 


19.  Evaluate      I  \xytdocdydz  for  all  positive  values  subject  to 
the  condition 


20.  Evaluate        I  A/—  dxdydz  for  all  positive  values  subject  to 
the  condition 

x  +  y  +  z  <  i.  \Tt. 

21.  Find  the  volume  of  a  cavity  just  large  enough  to  permit  of 
the  complete  revolution  of  a  circular  disk  of  radius  a,  whose  centre 
describes  a  circle  of  the  same  radius,  while  the  plane  of  the  disk 
is  constantly  parallel  to  a  fixed  plane  perpendicular  to  that  of  the 
circle  in  which  its  centre  moves.  iks(37r  +  8). 

22.  An  inverted  hollow  circular  cone  of  radius  a  and  height  h 
is    filled   with    material    of  which  the  density  varies  as  the  depth 
below  the  plane  of  the  base  ;  find  the  mass  of  the  material,  yu  being 
the  density  at  the  vertex.  -^-%n ^d'h. 

23.  A  vessel  consisting  of  a  hemisphere  of  radius  a  and  a  cylinder 
having  the  same  base  and  height  h  is  filled  with  material  having  the 
same  law  of  density  as  in  Ex.  22  ;  what  is  the  value  of  h  if  half  the 
mass  is  in  the  hemisphere  ?  (§  + 


I  ?2  GEOMETRICAL  APPLICATIONS.  [Art.   137. 

XL 

The  Polar  Element  of  Area. 

137.  When  polar  coordinates  are  used,  if  concentric  circles 
be  drawn,  corresponding  to  values  of  the  radius  vector  r  hav- 
ing the  common  difference  Ar, 
and  then  straight  lines  through 
the  pole  corresponding  to  values 
of  the  angular  coordinate  6  having 
the  uniform  difference  Ad,  any 
given  portion  of  the  plane  may  be 
divided,  as  in  Fig.  22,  into  small 
areas.  The  value  of  any  one  of 
these  is  readily  seen  to  be  Ar  .  rA6,  FIG.  22. 

where  r  has  a  value  midway  between  the  greatest  and  least 
values  of  r  in  the  area  in  question.     It  follows  that  the  result 


of  a  double  summation  of  this  element  between  proper  limits 
will  give  an  approximate  expression  for  the  given  area.  Hence 
the  double  integral 


A 


=   {(rdrdO, (i) 


which  (compare  Art.  122)  is  the  limit  of  this  expression  when 
Ar  and  Ad  are  both  diminished  without  limit,  is  the  exact  ex- 
pression for  the  given  area. 

138.  The  differential  expression  r  dr  dO,  or  polar  element  of 
area,  is  the  product  of  the  differentials  dr  and  rdO,  which  cor- 
respond to  the  mutually  rectangular  dimensions  Ar  and  rABoi 
the  element  of  summation.  These  factors  are  the  differentials 
of  the  motions  of  the  point  (r,  0),  respectively,  when  r  alone 


§  XL]  THE  POLAR  ELEMENT   OF  AREA. 


varies  and  when  0  alone  varies.  It  is  obvious  that  the  ele- 
ment of  area  for  any  system  of  coordinates  can  in  like  manner 
be  shown  to  be  the  product  of  the  corresponding  differentia] 
motions,  provided  only  that  these  motions  are  at  rigJit  angles 
to  one  another*  The  element  is  in  such  a  case  said  to  be 
ultimately  a  rectangle. 

139.  The  formula  used  in  section  VIII  is  in  fact  the  result 
of  performing  the  integration  for  r  in  formula  (i)  above.  Thus 
when  the  pole  is  outside  of  the  given  area,  as  in  Fig.  22,  we 
have 

A 


=  {  ^rdrdV  =  $  f  (r./  -  r*)dO, 


the  formula  of  Art.  HO.  The  limits  for  0  are  now  the  values 
which  make  r2  =  r1;  that  is,  in  the  case  of  a  continuous 
curve,  the  values  for  which  the  radius  vector  is  tangent  to  the 
curve.  So  also  when,  as  in  the  example  of  Art.  109,  the  pole 
is  on  the  curve  (so  that  rl  =  o  for  all  values  of  (f),  the  limits  for 
0  correspond  to  tangents  to  the  curve.  But,  when  the  pole  is 
within  the  curve,  we  assume  the  lower  limit  zero  to  avoid  nega- 
tive values  of  r,  and  then  make  6  vary  through  a  complete  revo- 
lution, that  is,  from  o  to  27T. 

140.  We  may,  of  course,  in  formula  (i)  integrate  first  for  6, 
the  limits  being  functions  of  r.     Thus 

A  = 

This  corresponds  to  summation  of  the  elements  along  a  cir- 
cular arc  of  radius  r,  so  as  to  form  the  element  (0a  —  O^rdr,  in 
which  the  angular  limits  are  two  consecutive  values  of  0  corre- 

*  In  other  words,  whenever  the  loci  of  constant  values  of  one  coordinate  cut 
orthogonally  the  like  loci  for  the  other  coordinate  ;  as  in  the  case  of  the  coordi- 
nates latitude  and  longitude  on  a  spherical  surface,  or  on  a  map  made  on  any 
system  in  which  the  representatives  of  the  parallels  and  meridians  cut  at  right 
angles. 


GEOMETRICAL   APPLICATIONS.  [Art.   140. 

spending  to  the  same  value  of  r,  and,  such  that  the  arc  lies 
within  the  given  area.  If  the  pole  is  outside  of  the  curve,  the 
limits  of  the  final  integration  will 'be  the  greatest  and  least 
values  of  r,  for  each  of  which  #2  —  0,  will  vanish  ;  but  if  the 
pole  is  within  the  curve,  the  arc  will  at  the  lower  limit  become 
a  complete  circumference,  and  the  integral  will  represent  the 
area  included  between  the  given  curve  and  this  circumference. 


Transformation  of  a  Double  Integral. 

|4f.  We  have  seen  in  Art.  126  that  a  double  integral  of  the 
general  form 


(i) 


may,  by  taking  x  and  y  as  rectangular  coordinates,  be  repre- 
sented by  the  volume  of  a  cylindrical  solid  whose  base  is  the 
area  in  the  plane  of  xy  determined  by  the  limits  of  integration, 
and  whose  upper  surface  is  defined  by  the  equation  z  =  <j)(x,y). 
If  we  introduce  the  new  independent  variables  r  and  6  con- 
nected with  x  and/  by  the  usual  polar  formulae, 

x  =  r  cos  6,        y  —  r  sin  6, 
it  is  obvious  that  the  same  solid  will  represent  the  integral 


II 


0(r  cos  6,  r  sin  0)rdrdQ (2) 


(in  which  the  element  of  area  dx  dy  is  replaced  by  the  polar 
element  r  dr  dO\  provided  the  integration  extends  over  the 
same  area,  of  which  the  boundary  must  of  course  now  be  ex- 
pressed  in  polar  coordinates. 

This  transformation  is  often  useful  when  the  equation  of 
the  boundary  is  simplified  by  the  use  of  polar  coordinates,  par- 


§  XL]    TRANSFORMATION   OF  A   DOUBLE  INTEGRAL,          1/5 

ticularly  when  0  is  also  simplified.     For  example,  the  last  of 
the  integrals  in  equation  (2),  Art.  136,  may  be  transformed  to 

7  f'P  ?      j   ja 

h        r*  .rdrdv  = 


f' 
\   \ 

JoJ 


where  the  limits  correspond  to  the  semicircle  whose  radius  is  a., 
142.  The  polar  element  of  area,  which  we  have  obtained 
geometrically  in  Art.  137,  may  also  be  derived  by  transforma- 
tion from  dx  dy,  by  the  method  given  below,  which  is  applica- 
ble to  any  transformation  of  variables. 

Since  the  element  depends  upon  the  independent  variation 
of  two  quantities,  it  is  necessary  to  replace  the  differentials, 
one  at  a  time,  by  those  of  the  new  variables  ;  the  other  vari- 
able at  each  stage  being  regarded  as  constant.  Let  us  first 
replace  dy  by  dd.  Expressing  y  in  terms  of  x  and  6,  by  the 
equations  of  transformation,  we  have 

y  =  x  tan  9  ; 
and,  differentiating  this,  x  being  regarded  as  constant, 

dy  =  x  sec2  9  d9. 

The  element  dx  dy  now  takes  the  form 

?  9d6. 


Again,  differentiating 

x  =  r  cos  6, 

0  being  now  regarded  as  constant,  we  have 

dx  =  cos  9  dr, 
and,  substituting  the  value  oixdx,  the  element  becomes  rdrdQ.* 

*  If  we  first  replace  dx  by  dr,  and  then  dy  by  dO,  we  shall  obtain  different 
expressions  for  the  differentials,  but  the  same  product.  But,  if  we  replace  dy 
by  dr,  and  then  dx  by  dB,  the  result  will  be  —  r  drdB.  The  negative  sign  is  due 
to  the  fact  that  the  mode  of  measuring  0  is  such  as  to  make  it  decrease  with  the 
increase  of  x,  when  y  is  positive. 


176  GEOMETRICAL   APPLICATIONS.  [Art.   143. 


Cylindrical  Coordinates. 

14-3.  In  determining  the  volume  of  a  solid,  it  is  sometimes 
convenient  to  use  the  polar  coordinates  r  and  6  in  place  of  x 
and  y,  while  retaining  the  third  coordinate  z  perpendicular  to 
the  plane  of  rd.  The  system  r,  6,  z  is  sometimes  called  that 
of  cylindrical  coordinates,  because  the  locus  of  r  =  a  constant 
is  the  surface  of  a  right  circular  cylinder.  The  element  of 
volume  in  this  system  is  obviously  the  product  of  dz  and  the 
polar  element  of  area,  that  is, 

rdr  d6  dz, 

which  has  the  ultimate  form  of  a  rectangular  parallelepiped. 

When  this  element  is  used  in  finding  the  volume  of  a  solid 
bounded  in  part  by  a  cylindrical  surface  of  which  the  elements 
are  parallel  to  the  axis  of  z,  like  that  represented  in  Fig.  20, 
Art.  126,  the  integration  for  z  is  naturally  performed  first,  its 
limits  being  the  values  of  z  in  terms  of  r  and  6  given  by  the 
equations  of  the  surfaces  which  cut  the  cylindrical  surface. 

14-4.  For  example,  let  us  find  the  volume  cut  from  a  sphere 
of  radius  a  by  a  right  circular  cylinder  having  a  radius  of  the 
sphere  for  one  of  its  diameters.  Taking  the  centre  of  the 
sphere  as  origin,  the  diameter  of  the  cylinder  as  initial  line, 
and  the  axis  of  z  parallel  to  the  axis  of  the  cylinder,  we  have, 
for  the  equation  of  the  sphere,  in  these  coordinates 

#  +  **  =  #, (i) 

and,  for  that  of  the  cylinder, 

r  =  a  cos  6 (2) 

The  limits  for  £  are  now  taken  from  equation  (i),  those  of  r 
are  zero  and  that  given  by  equation  (2),  and  those  of  #are  ±  £TT 


§  XL]  CYLINDRICAL    COORDINATES.  177 

which  make  r  in  equation  (2)  vanish.     Hence  we  have  for  the 
required  volume 


•-  «<t  cos  9  p  v(aa  —  r3) 

V=\d6\         rdr\ 

J  _-         J  »  -  4/(a*  —  r2) 

2 

7T 

.  -  /•  a  cos  8 

=  2[    d9\  (#- 

J  _"        J  o 


i  a  cos  6 


From  the  symmetry  of  the  solid  it  is  apparent  that  we  may 
take  the  limits  for  6  to  be  o  and  £TT,  and  double  the  result.* 
Thus 


- 
3  Jo 

14-5.  In  the  system  of  cylindrical  coordinates  z  and  r  may 
be  regarded  as  rectangular  coordinates  in  a  plane  which  passes 
through  the  axis  of  z,  and  makes  the  diedral  angle  6  with  a 
fixed  plane.  An  equation  between  z  and  r  independent  of  6  is 
thus  the  equation  of  a  surface  of  revolution  about  the  axis  of 
z,  and  at  the  same  time  it  is,  in  rectangular  coordinates,  the 
equation  of  the  generating  curve  (positive  values  of  r  only  being 
admissible).  The  volume  of  the  solid  of  revolution  is  therefore 

*  In  this  example,  the  result,  which  would  otherwise  have  been  written  in 

IT 

2#3  f2 

the  form  V  '  =•  —  I       (i  —  sin3  0X0,  would  not  have  been  correct,  because  the 
3   J.1 

2 

(a*  —  r2)'  which  occurs  in  the  integration  stands  for  the  positive  value  of  the 
radical,  whereas  in  the  fourth  quadrant  a  sin  9,  which  we  have  put  for  it,  stands 
for  the  negative  value. 


1/8  GEOMETRICAL    APPLICATIONS.  [Art.    145 

given  by  the  present  form  of  triple  integral  when  the  limits  of 
6  are  o  and  2/r,  and  the  ^-integration  extends  over  the  area 
of  the  generating  curve.  Thus 

f27r      f  f  f  f 

dO      rdrdz  =  2n\\  rdrdz, 


_  f: 


since  by  hypothesis  6  is  not  involved  in  the  limits  of  either  of 
the  other  variables  (compare  Art.  136). 

In  the  last  written  expression  for  V,  if  zero  is  the  lower 
limit  of  the  integration  for  r,  the  element  of  final  integration 
is  7tr*dz,  equivalent  to  the  lamina  of  Art.  1 16  ;  but,  if  we  inte- 
grate first  for  z,  we  have  for  the  final  integration  the  cylindrical 
element  2.n(z^  —  z^r  dr,  corresponding  to  the  method  employed 
in  Art.  120. 

Solids  of  Revolution  with  Polar  Coordinates. 

146.  The  volume  of  the  solid  produced  by  the  revolution 
about  the  initial  line  of  a  curve  given  in  polar  coordinates  may 
be  expressed  by  a  double  integral  of  which  the  element  is 
derived  from  the  polar  element  of 
area.  In  this  revolution,  every 
point  P  of  the  polar  element  of 
summation,  r  Ar  AQ,  constructed 
in  Fig.  23,  describes  a  circle  whose  FIG-  23- 

radius  is  PR  =  r  sin  0,  and  moves  always  in  a  direction  per- 
pendicular to  the  plane  of  the  element.  Therefore  the  volume 
described  by  the  element  is  equal  to  its  area  multiplied  by 
2Ttr  sin  6,  where  r  and  6  belong  to  some  point  within  the  ele- 
ment. Since  the  distinction  between  all  such  points  disap- 
pears at  the  limit,  we  have  for  the  element  of  volume 

27T7-2  sin  6 dr  dd  ; 

and  the  integral  of  this  expression  over  the  given  area  is  the 
required  volume. 


§  XL]  SOLIDS   OF  REVOLUTION.  1/9 

147.  For  example,  the  curve  in  Fig.  23  being  the  lemniscate 
r*  —  a2  cos  26, 

the  volume  generated  by  the  right-hand  loop,  or  rather  by  the 
half-loop  in  the  first  quadrant,  is 

F  =  2?r  p  pV sin  BdrdB  =  —  [ « (cos  2$)* sin  d dO. 

Transforming,  by  putting  x  =  cos  6,  since 
cos  26  =  2  cos2  O—i, 


3 
and  again,  by  putting  x  .\/2  =  sec  0, 

r,       yra3 1/2  f ;•  ,    , ,        Ttcf  A/2  t-  sin4  0  ,  , 

V=-  1 4 tan4  0  sec  0^/0=  —I4 —d<b. 

3      Jo  3      J0cos50 

Using  now  the  formula  of  reduction  (6),  Art.  70,  we  find 

fsin4  0        _      sin30         3  sin  0        3         i  +  sin  0 
J  cos5  0        ~  4  cos4  0  ~~  8  cos2  0      8  °g     cos  0 

whence,  applying  the  limits, 


=0,228  X   «3. 

J 


Polar  Coordinates  in  Space. 

148.  The  double  integral  employed  in  the  preceding  arti- 
cle is  in  fact  an  application  of  the  polar  system  of  coordinates 
in  space.  In  this  system,  a  point  is  determined  by  polar  coor- 
dinates in  a  plane  which  passes  through  a  fixed  axis  and  makes 
the  diedral  angle  0  with  a  fixed  plane  of  reference  passing 
through  the  axis.  In  Art.  146,  the  fixed  axis  is  the  initial  line 


i8o 


GEOMETRICAL   APPLICATIONS.  [Art.   148. 


OA,  and  taking  the  plane  of  the  paper  as  the  plane  of  refer- 
ence, the  third  coordinate  <p  is  the  angle  described  by  the 
upper  half-plane  in  the  revolution  of  the  figure.  The  differen- 
tial of  the  motion  of  P  when  0  varies  is  r  sin  B  d<p,  and  this 
motion  is  at  right  angles  to  the  plane  of  the  differentials  dr  and 
rdd\  hence  the  ultimate  element  of  volume  is  t2  sin  ddrdO  d(f>. 
But  in  the  case  of  the  solid  of  revolution  the  limits  of  the  0-in- 
tegration  are  O  and  2?r,  so  that  the  result  of  this  integration  is 
27i  r2  sin  QdrdQ,  the  element  of  the  double  integral  derived  in 
Art.  146. 

14-9.  In  the  equations  of  transformation  connecting  these 
coordinates  with  the  rectangular  system,  it  is  usual  to  take  the 
axis  of  z  as  that  about  which  the  angle 
<p  is  described,  and  the  plane  xz  as  that 
for  which  0  =  o.  Then  in  the  plane 
ZOR,  Fig.  24,  the  line  OR  is  usually 
taken  as  the  initial  line  for  the  polar 
coordinates  p  and  #,  so  that  the  three 
coordinates  of  P  are 

/    ^< 

R 


p=OP,      8 

Thus  p  in  this  system  is  the  distance  FIG.  24. 

of  the  point  P  from  a  fixed  pole,  and  8  and  0  correspond  to  the 
spherical  coordinates  latitude  and  longitude  on  a  sphere  whose 
radius  is  p.  Denote  the  radius  of  the  circle  of  latitude  BP  by 
r;  then 

r  =  p  cos  0, 

and  r  and  0  are  the  polar  coordinates  of  the  projection  of  P  in 
the  plane  of  xy.  Hence  the  formulae  connecting  the  rectangu- 
lar coordinates  x,  y,  z  with  p,  6  and  0  are 

x  =  r  cos  0  =  p  cos  8  cos  0, 
y  =  r  sin  0  =  p  cos  8  sin  0, 
z  =  p  sin  8. 


§  XL]  POLAR    COORDINATES  IN  SPACE.  l8l 

The  differentials  of  the  motions  of  Pwhen  one  of  the  coor- 
dinates p,  0,  and  0  varies,  the  other  two  remaining  constant, 
are,  respectively, 

dp,         pd6,         and         p  cos  Qd(f>\ 
hence  the  element  of  volume  is  their  product, 
p2  cos  6  dp  d6  dcf>, 

in  which  the  factor  cos  6  occurs,  in  place  of  the  factor  sin  6  which 
appears  in  the  element  derived  in  the  preceding  article,  because 
the  axis  of  rotation  is  now  perpendicular  to  the  initial  line. 

In  the  case  of  a  sphere  whose  centre  is  at  the  pole,  all  the 
limits  are  constant,  and  we  have 

=  f>^r:  cos—  f"--4^ 


Spherical  Coordinates. 

ISO.  If  we  give  to  p  the  constant  value  a,  6  and  0  become 
the  coordinates  latitude  and  longitude  of  a  point  upon  a  spheri- 
cal surface,  and  a  relation  between  6  and  0  becomes  the  equa- 
tion of  a  line  drawn  upon  the  spherical  surface.  For  example, 
a  formula  of  spherical  right  triangles  gives  for  the  equation  of 
the  great  circle  making  the  angle  A  with  the  equator  at  the 
zero  of  longitude 

sin  0=  cot  A  tan  0 (i) 

The  product  of  the  differentials  of  the  motions  of  P  corre- 
sponding to  variations  in  8  and  0,  Fig.  24,  that  is, 

a2  cos  6  dd  d(f>, 
is  the  element  of  spherical  surface.     Hence,  for  example,  to  find 


1  82  GEOMETRICAL   APPLICATIONS.  [Art.  150. 

the  triangular  area  included   between  the  equator  0  =  o,  the 
great  circle  equation  (i),  and  the  meridian  0  =  a,  we  have 


=  c?  f  a  f  cos  Odedtf>=  a2  ["sin 

»  o*  o  »  o 


where  6  is  the  function  of  (f>  defined  by  equation  (i).     From 
that  equation 

sin  0  sin  0 

sin  "  = 


4/(cot2  A  -f-  sin2  0)        ^/(cosec2  A  —  cos2  0)" 
Therefore 

ra  sin  0<^0  2-1  cos  0  "T 

S  =  aM    -— -  —. — -=—  arsm'1 

J  0  ^(cosec*1  ^4  —  cos*5  0)  cosec  ^4  J0 

=  a\A  —  sin~'(sin  A  cos  a)].* 

Volumes  in  general. 

151.  We  have  seen  in  Art.  123  that  the  boundary  of  the 
area  expressed  by  a  double  integral  may  consist  in  part  of  lines 
whose  equations  contain  only  one  of  the  variables,  namely, 
that  for  which  the  final  integration  takes  place.  But,  as  ex- 
plained in  Art.  125,  in  the  general  case,  it  is  necessary  to  regard 
the  given  area  as  made  up  of  positive  or  negative  parts  of  the 
kind  just  mentioned.  This  is  done  by  drawing  the  loci  of  con- 
stant values  of  the  final  variable  through  the  intersections  of 
the  lines  forming  the  boundary,  or  else  tangent  to  them  as  in 
Fig.  19.  These  parts  are  then  expressed  by  separate  integrals. 

*  If  B  is  the  other  oblique  angle  of  the  right  triangle,  another  trigonometric 
formula  gives 

cos  B  =  sin  A  cos  ex.  ' 

hence 

S  =  a\A  +  B  -  \it\, 

from  which  it  readily  follows  that  the  area  of  any  spherical  triangle  is  equal  to 
«*  X  the  spherical  excess  in  arcual  measure. 


§  XL]  VOLUMES  IN  GENERAL.  183 

So  also  we  have  seen  in  Art.  126  that  the  volume  expressed 
by  a  triple  integral  may  be  bounded  in  part  by  a  surface  whose 
equation  contains  only  two  of  the  variables,  namely,  those  for 
which  the  last  two  integrations  take  place.  But,  in  the  general 
case,  it  is  necessary  to  separate  the  volume  into  parts,  by  means 
of  such  surfaces  passed  through  the  lines  of  intersections  ot  the 
bounding  surfaces,  or  edges  of  the  given  solid. 

152.  Figs.  19  and  2O  illustrate  this  for  rectangular  coordi- 
nates, but  similar  considerations  apply  to  any  other  system,  and 
enable  us  to  decide  whether  it  is  possible  to  express  a  given 
volume  by  a  single  integral.  For  example,  let  it  be  required 
to  find  the  volume  common  to  the  solid  of  revolution  produced 
by  the  half-cardioid  OAB,  Fig.  25,  revolving  about  its  axis  OB, 
and  the  sphere  whose  centre  is  at  the  pole  and  whose  radius 
is  OC  =  c.  The  volume  is  the  result  of  integrating  the  element 

2zrr2  sin  6  dr  dB 

(found  as  in  Art.  146)  over  the  area  OA  C.  For  this  area  of 
integration,  6  =  7t  and  r  =  o  give  one  constant  limit  for  each 
variable,  and  the  others  are  determined  by  the  equation  of 

the  arc  OA  of  the  cardioid 

r  =  a(i  —  cos  0),  .     .     (i) 
and  that  of  the  circular  arc  AC, 

r  =  c.       ...     (2) 

Since  equation  (i)  contains  both  variables,  while  equation  (2) 
contains  r  only,  we  can,  by  integrating  first  for  0  (and  using 
equation  (i)  for  one  of  its  limits),  express  the  required  magni- 
tude by  a  single  integral  ;  thus, 


fVsi 


sin  BdBdr  =  211     i*(i  +  cos  B)dr. 


1  84  GEOMETRICAL   APPLICATIONS.  [Art.   152. 

Substituting  the  value  of  cos  0  from  equation  (i), 

V=  2 


If  in  this  example  we  integrate  first  for  r,  it  becomes  neces- 
sary to  find  0a,  the  value  of  d  for  the  point  of  intersection  A, 
and,  regarding  the  area  of  integration  as  separated  into  two 
parts  by  the  radius  vector  OA,  to  form  two  integrals,  in  one  of 
which  the  upper  limit  for  r  is  taken  from  equation  (i),  and  the 
limits  for  6  are  o*  and  Ol  •  while  in  the  other  r  is  taken  from 
equation  (2),  and  the  limits  for  0  are  Ol  and  TT. 

153.  As  a  further  illustration  of  these  principles,  let  us  re- 
sume the  consideration  of  the  volume  evaluated  in  Art.  14/5 
The  volume  is  completely  enclosed  by  the  spherical  and  cylin- 
drical surfaces  (i)  and  (2).  In  the  evaluation,  after  integrating 
for  z,  which  occurs  only  in  equation  (i),  the  whole  volume  was 
represented  by  a  double  integral;  the  limits  of  the  r#-integra- 
tion  were  determined  by  equation  (2)  regarded  as  representing 
an  area  in  the  plane  #  =  o.f  In  like  manner,  had  we  integrated 
first  for  6  which  occurs  only  in  equation  (2),  the  volume  would 
have  been  expressed  by  a  single  double-integral  expression. 

But  suppose  we  wished  to  perform  first  the  integration  for 
r  of  the  element 

r  dr  dO  dz. 

The  indefinite  integral  is  ^r2,  and  the  lower  limit  is  zero;  but 
the  upper  limit  is,  for  some  values  of  z  and  0,  given  by  equa 

*This  limit  also  is  in  fact  determined  by  the  intersection  of  two  sides  of  the 
area  of  integration,  namely,  that  of  the  curve  (i)  and  r  =  o  the  vanishing  inner 
edge  of  the  area. 

f  This  area  happened  to  be  entirely  within  the  "projection"  of  the  spher; 
(i)  ;  that  is,  the  circle  r  =  a,  within  which  only  z  in  equation  (i)  is  real.  Had 
this  not  been  the  case,  the  area  of  integration  would  have  been  only  that  com- 
mon to  the  curve  (2)  and  the  projection. 


§  XL]  VOLUMES  IN  GENERAL,  1  8$ 

tion  (l),  and  for  others  it  is  given  by  equation  (2).  In  other 
words,  the  extremity  of  rmoves  from  the  axis  of  z  outward  until 
it  reaches  either  the  spherical  or  the  cylindrical  surface.  Thus 
the  whole  field  of  ^-integration,  which  extends  from  z  =  —  a 
to  z  =  a,  and  from  6  =  —  \n  to  6  =  ^TT,  must  be  divided  into 
parts  corresponding  to  these  values  of  the  upper  limit  of  r.  Tak- 
ing, as  we  may  by  symmetry,  one-fourth  of  this  field  of  in- 
tegration for  one-fourth  of  the  volume  V,  the  dividing  line 
corresponds  to  the  intersection  of  the  sphere  and  cylinder;  it 
is  therefore  the  result  of  eliminating  r  between  equations  (i^ 
and  (2),  namely, 

z  =  a  sin  6. 

Now,  when  z  <  a  sin  6,  the  r  of  the  cylinder  is  less  than  that 
for  the  sphere,  and  is  therefore  to  be  taken  as  the  upper  limit 

in  J*3!  5  but,  when  ^  >  a  sin  0,  the  upper  limit  must  be  taken 
from  equation  (i).  Hence  \V  =  Fx  +•  F2*,  where 


and 

«sin0 


The  result  will  be  found  to  agree  with  that  of  Art.  144. 


Examples   XI. 

i.  Find,  by  integrating  rdr  d.6  first  with  respect  to  0,  the  area 
included  by  the  first  whorl  of  the  spiral  of  Archimedes 


and  the  initial  line;  also,  in  the  same  manner,  that  between  the  first 
and  second  whorls  and  the  initial  line.  i71"3^;  8?rV. 

*  The  surface  separating  Ft  and  Vi  is  that  of  a  right  conoid  generated  by 
a  line  passing  through  and  perpendicular  to  the  axis  of  2,  and  also  passing 
through  the  intersection  of  the  sphere  and  cylinder.  The  field  of  integration 
may  be  conceived  of  as  an  area  lying  upon  the  surface  of  the  cylinder  r  =  a, 
The  separating  surface  traces  upon  the  cylinder  the  line  z  =  a  sin  0. 


1  86  GEOMETRICAL   APPLICATIONS.  [Ex.  XI. 

2.  The  paraboloid  of  revolution 

x*  +  /  =  cz 

is  pierced  by  the  right  circular  cylinder 

x*  +  y  =  ax, 

whose  diameter  is  #,  and  whose  surface  contains  the  axis  of  the 
paraboloid;  find  the  volume  between  the  plane  of  xy  and  the  sur- 
faces of  the  paraboloid  and  of  the  .cylinder. 


32; 

3.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve 

2<Z37T  8#* 

r  =  a  cos  3  #.  ---  . 

3  9 

4.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve  , 

S  =  a*  cos'  6  +  F  sin3  6, 
supposing  b  <  a.  -----  («"  —  £')*. 

O  7 

5.  A  right  cone,  the  radius  of  whose  base  is  a  and  whose  alti- 
tude is  b,  is  pierced  by  a  cylinder  whose  base  is  a  circle  having  for 
diameter  a  radius  of  the  base  of  the  cone;  find  the  volume  common 

to  the  cone  and  the  cylinder.  be?  . 

-(9*  -  16). 

6.  The  axis  of  a  right  cone  whose  semi-vertical  angle  is  ot  coin- 
cides .with  a  diameter  of  the  sphere  whose  radius  is  0,  the  vertex 
being  on  the  surface  of  the  sphere  ;  find  the  volume  of  the  portion 
of  the  sphere  which  is  outside  of  the  cone.  4/r#3  cos4  a 

3 

7.  Find  the  volume  produced  by  the  revolution  of  the  lemniscate 

r1  =  a3  cos  26 
about  a  perpendicular  to  the  initial  line. 


§  XL]  EXAMPLES.  187 

8.  Find  the  volumes   generated  .by  the  revolution  of  the  large 
loop  and  by  one  of  the  small  loops  of  the  curve 

r  —  a  cos  &  cos  26 


about  a  perpendicular  to  the  initial  line. 


i6          5          32 

9.  What  is  the  volume  of  the  cavity  which  will  just  permit  the 
cardioid 

r  =  a(i  —  cos  6) 

to  rotate  about  a  line  on  its  plane  perpendicular  to  the  initial  line? 

16    +    57T 

—  7i 'a  . 
4 

10.  Find  the  whole  volume  enclosed  by  the  surface 

(x*  +  /  +  2s)3  =  a'xyz. 

Transform  to  the  coordinates  p,  0,  6,  and  show  that  the  solid  con 
sists  of  four  equal  detached  parts.  a3 

6' 

ii.   Find  the  volume  of  that  part  of  the  sphere 

x1  +  /  +  z*  —  a(x  +  y  +  z) 
for  which  all  three  coordinates  are  positive.  27f  +  *   3 

(/    • 

4 

12.  In  Ex.  ii  the  coordinate  planes  separate  the  sphere  into  the 
part  A  there  found,  three   adjacent   parts  B,  and  three   parts    C. 
Show   that    a   similar   integral    to    that    representing  A   represents 
B  —  C,  and  hence  derive  the  volumes  B  and  C. 

'  B  _  r  *     *  -  z"i ,.    c_  r  *     z71  -  ri  3 

|_44/3  12    J  [_4  4/3  I2     -T 

13.  Find  the  volume  generated  by  the  revolution  of  the  curve 


in  which  a  >  b,  about  the  axis  of  y. 


1 88  GEOMETRICAL    APPLICATIONS.  [Ex.  XI. 

Transform  to  polar  coordinates. 

nb(zb*  +  30°)  Tta4  _,  b 

~~6~          f  2  ^  -  £")  C          7i 

14.  Find  the  volume,  generated  by  the  curve  given  in  the  pre- 
ceding example  when  revolving  about  the  axis  of  x. 

7ta(2at  +  ^}  nb*  a+  <'a*  -  &') 


15.  Find  the  mass  of  a  circular  lamina  of  radius  a  if  the  density 
varies  as  the  distance  from  a  fixed  point  on  the  circumference  and 
is  /*  at  the  centre.  Interpret  the  integral  also  as  a  volume. 


9 

16.  Find  the  mass  of  a  square  lamina  of  side  20  if  the  density 
varies  as  the  distance  from  the  centre  and  is  f*  at  the  middle  of  a 
side.  4^r 

^-[|/2   +log(|/2    +    l)]. 

o 

17.  Find    the  volume    between    the    surface    generated    by  the 
revolution  of  the  cardioid 

r  =  a(i  —  cos  B) 

about  the  initial  line  and  the  plane  which  touches  this  surface  in 
circle.  no1 

192 

1  8.  Transform  the  triple  integral  element  dx  dy  dz  into  the  polar 
element  p"  cos  6  dp  d&  d<p.     See  Arts.  142  and  149. 

19.  Find  the  separate  values  of  V^  and  V^  in  Art.  153. 


20.  Find  the  volume  cut  from  a  right  circular  cone  of  radius  a 
and  height  a  by  a  plane  passing  through  the  centre  of  the  base  and 
parallel  to  an  element  of  the  cone.  3^  —  4  „ 

~~~ 


XII.] 


RECTIFICATION  OF  PLANE    CURVES. 


189 


XII. 
Rectification  of  Plane  Curves. 

154.  A  curve  is  said  to  be  rectified  when  its  length  is  found 
in  terms  of  that  of  some  given  straight  line. 

It  is  shown  in  Diff.  Calc.  that,  if  s  denotes  the  arc  of  a  curve 
given  in  rectangular  coordinates, 

ds  —  ^(dx*  +  dy*). 

If,  in  this  expression,  the  value  of  dy  in  terms  of  x  and  dx 
be  substituted,  the  length  of  an  arc  in  which  x  varies  continu- 
ously is  found  by  integration,  the  limits  being  the  values  of  x 
at  the  extremities. 

Thus,  in  the  case  of  the  semi-cubical  parabola  ay*  =  x3, 

-\\fxdx  \/(QX  +  4a)  , 

dy  =    v         ;     whence     ds  =  -         ,         dx. 

1*0  find  the  arc  between  the  origin  (a  point  of  the  curve)  and 
the  point  (x,  y),  we  integrate  between  zero  and  x.     Thus 
if  I  Sa 

~  24/#J0  ~  27^0,  9  27  ' 

155.  When  x  and  y  are  most  conveniently  expressed  as 
functions  of  a  third  variable,  the  expression  for  ds  in  terms  of 
this  variable  may  be  used.    For  example,  in  the  case  of  the  curve 

A 


. 

7  '=  ' 
b1 


we  put,  as  in  Art.  107, 
x  —  a  sin3  0, 
y  =  b  cos3  ip  ; 
whence 

dx  =    a  sin2     cos 


dy  =  — 


FlG  26. 
cos2  fi  sin 


igO  GEOMETRICAL   APPLICATIONS.  [Art.   155. 

and  therefore 

ds  =  3  sin  ^  cos  ^  d$  |/(«2  sin2  ^  +  ^  cos2  ^) 

=  f  t/[(>2  -  J8)  sin2  0 


The  points  A  and  B,  Fig.  26,  correspond  to   if>  —  o   and 
i[>  =  \7t  respectively;  hence,  integrating,  we  have 

7T 
A      T~»  It*        O  *  *  t         l£/          I          L-'         \_  V/ .~)        V^     /  "        I  •*  C'fr  f  *-*•  \          (IU 

arc  y4j5  = 


o       a   - 


Change  of  the  Sign  of  ds. 

156.  In  the  example  above,  x  and^j/  being  one-valued  func- 
tions of  ?/>,  every  value  of  ip  corresponds  to  a  definite  point  of 
the  curve  ;  and,  as  »/>  varies  from  o  to  2?r,  the  point  (x,  y)  de- 
scribes the  whole  curve  in  the  direction  ABCD.     But  it  will  be 
noticed  that,  dtf>  remaining  positive,  the  value  of  ds  becomes 
zero  and  changes  sign  when  ^  passes  through  either  of  the  val- 
ues o,  I-TT,  TT,  or  f  n.     This  corresponds  geometrically  to  the  fact 
that  the  point  (x,  y)  stops  and  reverses  the  direction  of  its  mo- 
tion, forming  a  stationary  point  or  ctisp  at  each  of  the  points  AT 
B,  C  and  D,  as  shown  in  Fig.  26.     Such  points  are  thus  indi- 
cated by  a  change  of  sign   in  ds,  and  the  arcs  between  them 
must  be  considered  separately,  because  the  corresponding  def- 
inite integrals  have  different  signs. 

Polar  Coordinates. 

157.  It  is  proved  in  Diff.  Calc.  that  in  polar  coordinates 

This  is  usually  expressed  in  terms  of  9.     For  example,  in  the 
case  of  the  cardioid  r  =  a(\  —  cos  6}  =  2a  sin2  £0,  we  have 

dr  =  2a  sin  £#  cos  %6  d9, 
whence,  by  substitution, 

ds  =  2a  sin  £0  dd. 


§  XII.]  RECTIFICATION  OF  CURVES.  19 r 

The  limits  for  the  whole  perimeter  of  the  curve  are  o  and  2n, 
and  ds  remains  positive  for  the  whole  interval.     Therefore 

f2ir  0  0~\21r 

s  =  2a\    sin  —  dO  =  —  Aa  cos  -       =  Sa. 

Jo  2  2J0 


Rectification  of  Curves  of  Double  Curvature. 

158.  Let  a  denote  the  length  of  the  arc  of  a  curve  of  double. 
curvature  ;  that  is,  one  which  does  not  lie  in  a  plane,  and  sup- 
pose the  curve  to  be  referred  to  rectangular  coordinates  x,  y 
and  z.  If  at  any  point  of  the  curve  the  differentials  of  the 
coordinates  be  drawn  in  the  directions  of  their  respective  axes, 
a  rectangular  parallelepiped  will  be  formed,  whose  sides  are 
dx,  dy  and  dz,  and  whose  diagonal  is  da.  Hence 

da  =  V(dx*  +  df  +  d£). 

The  curve  is  determined  by  means  of  two  equations  connect- 
ing x,  y  and  z,  one  of  which  usually  expresses  the  value  of  y  in 
terms  of  x,  and  the  other  that  of  z  in  terms  of  x.  We  can 
then  express  da  in  terms  of  x  and  dx. 

If  the  given  equations  contain  all  the  variables,  equations 
of  the  required  form  may  be  obtained  by  elimination. 

159.  An  equation  containing  the  two  variables  x  and  y 
only  is  evidently  the  equation  of  the  projection  upon  the  plane 
of  xy  of  a  curve  traced  upon  the  surface  determined  by  the 
other  equation.  Let  s  denote  the  length  of  this  projection  : 
then,  since  d&  =  dx*  +  dy*, 


in  which  d±  may,  if  convenient,  be   expressed  in  polar  coordin- 

ates ;  thus, 


192  GEOMETRICAL  APPLICATIONS.  [Art.  l6o. 

160.  As  an  illustration,  let  us  use  this  formula  to  deter- 
mine the  length  of  the  loxodromic  curve  from  the  equation  of 
the  sphere, 

x2+<f  +  t?  =  a*,     .......    (i) 

upon  which  it  is  traced,  and  its  projection  upon  the  plane  of 
the  equator,  of  which  the  equation  is 


or  in  polar  coordinates 

2a  =  r  (ene  +  c-"e)  .......     (2) 

Equation  (i)  is  equivalent  to 

f»  +  *>  =  «*; 

and,  denoting   the  latitude  of  the  projected  point  by  <f>,  this 

gives 

z  =  a  sin  ^,  r  —  a  cos  <f>.     .     .     .     (3) 

In  order  to  express  dB  in  terms  of  <f>,  we  substitute  the  value 
of  r  in  (2)j  whence 

e»6  4.   e-*6  —  2  sec  ^       ......       (4) 

and  by  differentiation 

e"°  -  e-»»  =  -  sec  ^  tan  ^  -=?  .....     (5) 

w  «cr 

Squaring  and  subtracting, 


which  reduces  to 


I 

§  XII.]          LI'.NGTH  OF   THE  LOXODROMIC  CURVE.  1 93 

From  equations  (3)  and  (6) 


dr*  =  c?  sin2 
dz*  —  a2  cos2 

whence  substituting  in  the  value  of  da  (p.  191) 

da  =  a  V  ( i  +-s)dfa 

\        n2/ 

Integrating, 

(3  =  a  — d(b  =  a  — •  ( (3  —  a). 

n         ja  n 

where  a  and  fi  denote    the    latitudes    of    the    extremities   of 
the  arc. 

Examples  XII. 

i.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 
catenary 


and  show  that  the  area  between  the  coordinate  axes  and  any  arc  is 
proportional  to  the  arc. 

X 

c   (  7       ' 
s  —  -  (e    —  e 
2  \ 

A  —  cs. 

2.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 
paraooia 

y*  =  4ax. 

.,  n  Vx  +  V(x  +  a) 

t/(ax  +  x)  -I-  aloe—  —  . 

Va        >    • 


194  GEOMETRICAL   APPLICATIONS.  [Ex.  XII. 

3.  Find  the  length  of  the  curve 


between  the  points  whose  abscissas  are  a  and  b. 


4.  Find  the  length,  measured  from  the  origin,  of  the  curve 

a   -x* 
y  =  a  log  — 5 — . 

a  +  x 
a  log x. 

a  —  x 

5.  Given  the  differential  equation  of  the  tractrix, 

dy__  _         y 

dx  V(&*  —  y*} ' 

and,  assuming  (o,  a)  to  be  a  point  of  the  curve,  find  the  value  of  s  as 
measured  from  this  point,  and  also  the  value  of  x  in  terms  of  y  ;  that 
is,  find  the  rectangular  equation  of  the  curve. 

y 

s  =  a  log-. 

.      a  +  v(a*  —  y2)         ,/  *        ^ 
x  =  a  log  — 


6.  Find  the  length  of  one  branch  of  the  cycloid 

x  =  a  (ip  —  sin  ^),  y  =  a  (i  —  cos  ^). 

Sa. 

•j.  When  the  cycloid  is  referred  to  its  vertex,  the  equations  being 

x  —  a  (i  —  cos  ^),  y  =  a  ($  +  sin  ^), 

prove  that  s  = 


§  XII.]  EXAMPLES.  195 

8.  Find  the  length  from  the  point  (a,  o)  of  the  curve 

x  =  20,  cos  y>  —  a  cos  21/1, 

y  =  20,  sin  t/>  —  a  sin  2ip. 

8a  (i  —  cos£  ^). 

9.  Show  that  the  curve, 

x  —  3  a  cos  ^  —  20,  cos*  ^,  y=  20.  sin3  ^ 

has  cusps  at  the  points  given  by  ^  =  o  and  ^  —  ?r ;  and  find  the 
whole  length  of  the  curve.  120. 

10.  Find  the  length  of  the  arc  of  the  parabola 

(-)'+($'=' 

\0  /        \  bl 
between  the  points  where  it  touches  the  axes. 

a3  +  bs            (fb*              \_V(a*  +  &*)  +  a~\  [V(^*  +  &*}  +  ^1 
-t- ^  losr 

I     i      rt      •     «    •  To\*        o 


ab 
ii.  Show  that  the  curve 

x  =  20,  cos2  B  (3  —  2  cos4  #),  y  =  40  sin  6  cos3  Q 


o 

has  three  cusps,  and  that  the  length  of  each  branch  is  —  . 


12.  Find  the  length  of  the  arc  between  the  points  at  which   the 
curve 

x  =  a  cos9  #  cos  20,  y  =  a  sin5  0  sin  26 

2+^2 

cuts  the  axes.  -  a. 


196  GEOMETRICAL  APPLICATIONS.  [Ex.  XII. 

13.  Show  that  the  curve 

x  =  a  cos  ip  (i  +  sin3  rf>), 
y  =  a  sin  ^  cos2  ip 

is  symmetrical  to  the  axes,  and  find  the  length  of  the  arcs  between 

the  cusps.  /  i 

2  —  sin-1 


a  (  4/2  +  cos-1  — 


14.  Find  the  length  of  one  branch  of  the  epicycloid 

a  +  b  . 


x  =  (a  +  ft)  cos  ip  —  b  cos 
y  =  (a  +  b)  sin  ip  —  b  sin 


, 

ip. 

U  (a  +  6) 

a 
15.  Show  that  the  curve 

x  —  pa  sin  ?/'  —  4<z  sin3  ^>, 
y  =  —  30  cos  if?  +  4.a  cos3  ?/? 

is  symmetrical  to  the  axes,  and  has  double  points  and  cusps  :  find  the 
lengths  of  the  arcs,  O)  between  the  double  points,  (/?)  between  a 
double  point  and  a  cusp,  and  (y)  the  arc  connecting  two  cusps,  and  not 
passing  through  the  double  points. 

(a),  a(7t  +  31/3); 


16.  Find  the  whole  length  of  the  curve 


x  —  30  sn  0  — 
=  a  cos'     . 


§  XII.]  EXAMPLES.  197 

17.  Find  the  length,  measured  from  the  pole,  of   any  arc  of  the 
equiangular  spiral 

r  =  as"9, 
in  which  n  =  cot  a.  r  sec  a. 

1 8.  Prove  by  integration  that  the  arc  subtending  the  angle  0  at  the 
circumference  in  a  circle  whose  radius  is  a,  is  206. 

19.  Find  the  length,  measured  from  the  origin,  of  the  curve  defined 
by  the  equations 


20.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  surfaces 

y  —  4«  sin  x,  z  =  21?  (2x  +  sin  2x). 

(472"  +  i)x  +  20*  sin  2X. 

21.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  cylindrical  surfaces 

(y  —  *Y  =  4<zx,  ga  (z  —  A')*  =  4^'. 


22.  If  upon  the  hyperbolic  cylinder 

/_!'- 
c*       b*         ' 

a  curve  whose  projection  upon  the  plane  of  xy  is  the  catenary 

-r  -r 

^    /    c     i  c  \ 

y  =  -  (e   +  t     ) 

be  traced,  prove  that  any  arc  of  the  curve  bears  to  the  corresponding 
arc  of  its  projection  the  constant  ratio  V(b*  +  t*)  '•  c. 


198  GEOMETRICAL   APPLICATIONS.  [Art.    161. 

XIII. 

Surfaces  of  Solids  of  Revolution. 

i6l.  The  surface  of  a  solid  of  revolution  may  be  generated 
by  the  circumference  of  the  circular  section  made  by  a  plane 

perpendicular  to  the  axis  of  revolu- 
tion. Thus  in  Fig.  27,  the  surface 
produced  by  the  revolution  of  the 
curve  AB  about  the  axis  of  x  is  re- 
garded as  generated  by  the  circum- 
ference PQ.  The  radius  of  this  cir- 
^umference  is  y,  and  its  plane  has  a 
motion  whose  differential  is  dx,  but 
every  point  in  the  circumference  itself 
has  a  motion  whose  differential  is  ds,  s 
denoting  an  arc  of  the  curve  AB. 
Hence,  denoting  the  required  surface  by  S,  we  have 

dS  —  2ny  ds  =•  27ty  \f(dx*  +  dy*}. 

The  value  of  dS  must  of  course  be  expressed  in  terms  of  a  single 
variable  before  integration. 

162.  For  example,  let  us  determine  the  area  of  the  zone  of 
spherical  surface  included  between  any  two  parallel  planes. 
The  radius  of  the  sphere  being  a,  the  equation  of  the  revolv- 
ing curve  is 

x*  +  y2  —  az; 

whence  y  =    V(a2  —  x2), 

x  dx 
dy  — 77-0 -or, 

^  A/ 1  XV*  -V*  \ 


dx 


and  dS  =  2na  dx\ 


§  XIII.]       SURFACES    OF   SOLIDS    OF  REVOLUTION.  199 

therefore 

5  =  2-na  \dx  =  2na  (x2  —  x^) . 

Since  x^  —  x±  is  the  distance  between  the  parallel  planes, 
the  area  of  a  zone  is  the  product  of  its  altitude  by  2na,  the 
circumference  of  a  great  circle,  and  the  area  of  the  whole  sur- 
face of  the  sphere  is  4^a2. 

163.  When  the  curve  is  given  in  polar  coordinates,  it  is  con- 
venient to  transform  the  expression  for  ,5"  to  polar  coordinates. 
Thus,  if  the  curve  revolves  about  the  initial  line, 


S  =  2n \y  ds  =  27t\r  sin  6 ^(dr*1  + 
For  example,  if  the  curve  is  the  cardioid 


we  find,  as  in  Art. 


r  =2a  sin*—  v  , 

2 


ds  =  2a  sin  —  6  dd. 

2 


Hence 

5  =  I6**8  f  "sin4  -  0  cos  -  6  dB 

Jo          2  2 


sm°—  c/      = 


5  2    Jo          5 


Areas  .of  Surfaces  in   General. 

1 64-.  Let  a  surface  be  referred  to  rectangular  coordinates  x, 
y  and  z ;  the  projection  of  a  given  portion  of  the  surface  upon 
the  plane  of  xy  is  a  plane  area  determined  by  a  given  relation 
between  x  and  y.  We  may  take  as  the  elements  of  the  surface 
the  portions  which  are  projected  upon  the  corresponding 


200 


GEOMETRICAL   APPLICATIONS. 


[Art.  164. 


elements  of  area  in  the  plane  of  xy.  If  at  a  point  within  the 
element  of  surface,  which  is  projected  upon  a  given  element 
Ax  Ay,  a  tangent  plane  be  passed,  and  if  y  denote  the  inclina- 
tion of  this  plane  to  the  plane  of  xy,  the  area  of  the  correspond- 
ing element  in  the  tangent  plane  is 

sec  y  Ax  Ay. 

The  surface  is  evidently  the  limit  of  the  sum  of  the  elements 
in  the  tangent  planes  when  Ax  and  Ay  are  indefinitely  dimin- 
ished. Now  sec  Y  is  a  function  of  the  coordinates  of  the  point 
of  contact  of  the  tangent  plane  ;  and  since  these  coordinates 
are  values  of  x  and  y  which  lie  respectively  between  x  and 
x  4-  Ax  and  between  y  and  y  -+-  Ay,  it  follows,  as  in  Art.  122, 
that  this  limit  is 


5  =      sec  Y  dx  dy\ 

165,  The  value  of  sec  Y   may  be   derived  by   the  following 

method.  Through  the  point  P  of 
the  surface  let  planes  be  passed 
parallel  to  the  coordinate  planes, 
and  let  PD,  and  PE,  Fig.  28,  be  the 
intersections  of  the  tangent  plane 
with  the  planes  parallel  to  the 
planes  of  xz  and/^r.  Then  PD  and 
PE  are  tangents  at  P  to  the  sec- 
tions of  the  surface  made  by  these 
planes.  The  equations  of  these 
sections  are  found  by  regarding  y 

and  x  in  turn  as  constants  in  the  equation  of  the  surface  ;  there- 
fore, denoting  the  inclinations  of  these  tangent  lines  to  the  plane 
of  xy  by  (j>  and  ^>,  we  have 


FIG.  28. 


dz 


and 


tan  i/}  = 


dz 
dy> 


§  XIII.]  AREAS  OF  SURFACES  IN  GENERAL.  2OI 

in  which  —  and-r-  are  partial  derivatives  derived  from  the  equa- 
dx        ay 

tion  of  the  surface. 

If  the  planes  be  intersected  by  a  spherical  surface  whose 
centre  is  P,  ADE  is  a  spherical  triangle  right  angled  at  A, 
whose  sides  are  the  complements  of  ^  and  ip.  Moreover,  if  a 
plane  perpendicular  to  the  tangent  plane  FED  be  passed 
through  AP,  the  angle  FPG  will  be  y,  and  the  perpendicular 
from  the  right  angle  to  the  base  of  the  triangle  the  comple- 
ment of  y. 

Denoting  the  angle  EAF  by  6,  the  formulae  for  solving 
spherical  right  triangles  give 


tan  if}  .     a      tan  $ 

cos  6  —  -  £•  ,  and  sin  0  =  - 

tan  y  tan  y 


Squaring  and  adding, 

tan2  if}  +  tan2 


__ 


tan2  y  —  tan2  if}  +  tan2  <j>  ; 


whence  sec2  y  —  I  +     -7-  -7-     • 

dxj         \dyj 


Substituting  in  the  formula  derived  in  Art.    164  ,  we  have 


166.  It  is  sometimes  more  convenient  to  employ  the  polar 


2O2  GEOMETRICAL   APPLICATIONS.  [Art.  1  66. 

element  of  the  projected  area.     Thus  the  formula  becomes 
5  =      sec  yr  dr  dd, 

where  sec  y  has  the  same  meaning  as  before. 

For  example,  let  it  be  required  to  find  the  area  of  the  sur- 
face of  a  hemisphere  intercepted  by  a  right  cylinder  having  a 
radius  of  the  hemisphere  for  one  of  its  diameters.  From  the 
equation  of  the  sphere, 


s?=rf,     .......     (i) 

we  derive 

dz  _       x  dz  _      y 

dx~       z*  dy~       2  ' 


whence 


/f  ^ 

sec  7=4/1+       -r     + 
¥  L 


(dz\^~\      a 

(-r)     =  — 
dxj        \dy/  J      z 


(  c         [(rdrdB 

therefore  o  =  a\\  -  -, 

the  integration  extending  over  the  area  of  the  circle 

r  =  a  cos  d  .......     ,      (2) 

Since  equation  (i)  is  equivalent  to 


=  a 


§  XIII.]  AREAS  OF  SURFACES  IN  GENERAL.  203 

From  (2)  the  limits  for  r  are  r^  =  0,  and  r2  =  a  cos  #, 
hence 


in  which  a  sin  6  is  put  for  \h& positive  quantity  V(a*  —  r.?}.  The 
limits  for  6  are  —  \n  and  \n,  but  since  sin  6  is  in  this  case  to 
be  regarded  as  invariable  in  sign,  we  must  write 


71 

5  =  2a*  [  2(i  -  sin  0)  dB  =  naz  -  2a\ 

Jo 


If  another  cylinder  be  constructed,  having  the  opposite  radius 
of  the  hemisphere  for  diameter,  the  surface  removed  is 
2/rtf2  —  4«2,  and  the  surface  which  remains  is  4*1*,  a  quantity 
commensurable  with  the  square  of  the  radius.  This  problem 
was  proposed  in  1692,  in  the  form  of  an  enigma,  by  Viviani,  a 
Florentine  mathematician. 


Examples  XIII. 

i.  Find  the  surface  of  the  paraboloid  whose  altitude  is  a,  and  the 
radius  of  whose  base  is  b. 

(*  +  *•)*-*]. 


2.  Prove  that  the  surface  generated  by  the  arc  of  the  catenary  given 
in  Ex.  XII.,  i,  revolving  about  the  axis  of  xt  is  equal  to 

n(cx  +  sy). 

3.  Find  the  whole  surface  of  the  oblate  spheroid  produced  by  the 


2O4  GEOMETRICAL  APPLICATIONS.  [Ex.  XIII. 

revolution  of  an  ellipse  about  its  minor  axis,  a  denoting  the  major, 
b  the  minor  semi-axis,  and  e  the  excentricity,  —  —     — 


27ta   +  n  -  log 
6 


e         i  —  e 


4.  Find  the  whole  surface  of  the  prolate  spheroid  produced  by  the 
revolution  of  the  ellipse  about  its  major  axis,  using  the  same  notation 
as  in  Ex.  3. 


2/T^2  +  27tab  . 

e 

5.  Find  the  surface  generated  by  the  cycloid 

x  =  a  (^  —  sin  ^),         y  =  a  (i  —  cos^) 

revolving  about  its  base.  —  na*. 

3 

6.  Find  the  surface  generated  when  the  cycloid  revolves  about  the 
tangent  at  its  vertex. 


7.  Find  the  surface  generated  when  the  cycloid  revolves  about  its 
axis. 

_  4 
3 


8.  Find  the  surface  generated  by  the  revolution  of  one  branch  of 
the  tractrix  (see  Ex.  XII.,  5)  about  its  asymptote. 

2-n  a*. 


§X!!I]  EXAMPLES.  205 

9.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 
x  of  the  portion  of  the  curve 


which  is  on  the  left  of  the  axis  of  y. 

7t[\/2    +  log   (l    +     V'2)]. 

10.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 

of  the  arc  between   the  points  for  which  x  =•  a  and  x  =  b  in  the 
i  _  i  _ 


x 
hyperbola 

xy  =  tf 


Tlk 


r     i>" 

L    g<? 


ii.  Show  that  the  surface  of  a  cylinder  whose  generating  lines  are 
parallel  to  the  axis  of  z  is  represented  by  the  integral 


S  =  \z  ds, 

where  s  denotes  the  arc  of  the  base  in  the  plane  of  xy.  Hence, 
deduce  the  surface  cut  from  a  right  circular  cylinder  whose  radius  is 
a,  by  a  plane  passing  through  the  centre  and  making  the  angle  at  with 
the  plane  of  the  base.  20*  tan  a. 

12.  Find  the  surface  of  that  portion  of  the  cylinder  in  the  problem 
solved  in  Art.  1 66,  which  is  within  the  hemisphere.  20?. 

13.  Find  the  surface  of  a  circular  spindle,  a  being  the  radius  and 
zc  the  chord. 


c  —  4/(tf3  —  ^"jsin-1- 


206  GEOMETRICAL   APPLICATIONS.  [Art.    167. 


XIV. 

The  Area  generated  by  a  Straight  Line  moving  in  any 
Manner  in  a  Plane. 

167.  If  a  straight  line  of  indefinite  length  moves  in  any  man- 
ner whatever  in  a  plane,  there  is  at  each  instant  a  point  of  the 
line  about  which  it  may  be  regarded  as  rotating.     This  point  we 
shall  call  the  centre  of  rotation  for  the  instant.     The  rate  of 
motion    of   every   point    of    the   line  in  a  direction  perpendic- 
ular to  the  line  itself  is  at   the   instant  the  same  as  it  would 
be  if  the  line  were  rotating  at  the  same  angular  rate  about  this 
point    as    a    fixed   centre.*     Hence  it   follows  that    the    area 
generated  by  a  definite  portion  of  the  line  has  at  the  instant 
the  same  rate  as  if  the  line  were  rotating  about  a  fixed  instead 
of  a  variable  centre. 

168.  Suppose  at  first  that   the  centre  of  rotation  is  on  the 
generating  line  produced,  pl  and  fo  denoting  the  distances  from 
the  centre  of  the  extremities  of  the  generating  line,  and  let  $ 
denote  its  inclination  to  a  fixed  line.     By  substitution  in  the 
general  formula  derived  in  Art.  no,  we  have 


dA  =  - 


*  Compare  Diff.  Calc.,  Art.  326  [Abridged  Ed.,  Art.  176],  where  the  moving 
line  is  the  normal  to  a  given  curve,  and  the  centre  of  rotation  is  the  centre  of  cur- 
vature of  the  given  curve.  If  the  line  is  moving  without  change  of  direction,  the 
centre  is  of  course  at  an  infinite  distance. 

When  the  line  is  regarded  as  forming  a  part  of  a  rigidly  connected  system  in 
motion,  its  centre  of  rotation  is  the  foot  of  a  perpendicular  dropped  upon  it  from 
the  instantaneous  centre  of  the  motion  of  the  system.  Thus,  if  the  tangent  and 
normal  in  the  illustration  cited  are  rigidly  connected,  the  centre  of  curvature,  C,  is 
the  instantaneous  centre  of  the  motion  of  the  system,  and  the  point  of  contact,  P, 
is  the  centre  of  rotation  for  the  tangent  line. 


§  XIV.]        AREAS  GENERATED  BY  MOVING  LINES.  2O7 


Applications. 

169.  The  area  between   a  curve  and   its  evolute   may  be 
generated  by  the  radius  of  curvature  p,  whose  inclination   to 
the  axis    of  x  is  <j>  +  $TT,  in  which   $  denotes  the    inclination 
of   the  tangent  line.      Since   the    centre    of    rotation    is    one 
extremity  of  the  generating   linep,  the  differential  of  this  area 
is  found  by  substituting  in  the  general  expression  p1  =  o  and 
/?2  =p.     Hence  when  p  is  expressed  in  terms  of  </>, 

A  =  l- 

2 

expresses  the  area  between  an  arc  of  a  given  curve,  its  evolute, 
and  the  radii  of  curvature  of  its  extremities,  the  limits  being 
the  values  of  <j)  at  the  ends  of  the  given  arc. 

170.  For  example,  in  the  case  of  the  cardioid 

r  —  a(i  —  cos  6), 

it  is  readily  shown,  from  the  results  obtained  in  Art.  157,  that 
the  angle  between  the  tangent  and  the  radius  vector  is  \6;  and 
therefore  <j)  =  J-#,  and 

ds      Aa   .    6 
p  =  — .  —  —  sin  -  . 
•  d<f>       3        3 

To  obtain  the  whole  area  between  the  curve  and  its  evolute, 
the  limits  for  8  are  o  and  2n  ;  hence  the  limits  for  </>  are  o 
and  3?r.  Therefore 


A     !  f317^,^     s^r3*  •  2<t> 

A  =  -,    f?  d<p  =  -        sin2  — 
2Jo  9  Jo        3 


4.7TO? 


171.  As    another   application    of    the    general    formula    of 
Art.  1 68,  let   one  end   of  a  line  of  fixed  length  a  be  moved 


208  GEOMETRICAL   APPLICATIONS.  [Art.    I/I. 

along  a  given  line  in  a  horizontal  plane,  while  a  weight  at- 
tached to  the  other  extremity  is  drawn  over  the  plane  by  the 
line,  and  is  therefore  always  moving  in  the  direction  of  the 
line  itself.  The  line  of  fixed  length  in  this  case  turns  about 
the  weight  as  a  moving  centre  of  rotation.  Hence  the  area 
generated  while  the  line  turns  through  a  given  angle  is  the 
same  as  that  of  the  corresponding  sector  of  a  circle  whose 
radius  is  a. 

The  curve  described  by  the  weight  is  called  a  tractrix,  and 
the  line  along  which  the  other  extremity  is  moved  is  the  direc- 
trix. When  the  axis  of  x  is  the  directrix,  and  the  weight 
starts  from  the  point  (o,  a),  the  common  tractrix  is  described ; 
hence  the  area  between  this  curve  and  the  axes  is  ^nd1. 

172.  Again,  in  the  generation  of  the  cycloid,  Diff.  Calc., 
Art.  278  [Abridged  Ed.,  Art.  156],  the  variable  chord  RP  may 
be  regarded  as  generating  the  area.  The  point  R  has  a  motion 
in  the  direction  of  the  tangent  RX;  the  point  P  partakes  of 
this  motion,  which  is  the  motion  of  the  centre  C,  and  also  has 
an  equal  motion,  due  to  the  rotation  of  the  circle  in  the  direc- 
tion of  the  tangent  to  the  circle  at  P.  Since  the  tangents 
at  P  and  R  are  equally  inclined  to  PR,  the  motion  of  P  in  a 
direction  perpendicular  to  PR  is  double  the  component,  in  this, 
direction,  of  the  motion  of  R.  Therefore  the  centre  of  rota- 
tion of  PR  is  beyond  R  at  a  distance  from  it  equal  to  PR. 
Hence,  denoting  PRO  by  ^, 

/>!  —  PR  =  2a  sin  <j),  pz  =  2PR  —  4/2  sin  <f>. 

Substituting  in  the  formula  of  Art.  168,  we  have  for  the  area 
of  the  cycloid,  since  PRO  varies  from  o  to  ?r, 


A  =  6a2      sin2  $  d$  =  37^. 

Jo 


§  XIV.] 


SIGN  OF   THE   GENERATED  AREA. 


209 


Sign  of  the  Generated  Area. 

173.   Let  AB  be  the  generating  line,  and  £7  the  centre  of 
rotation.     The  expression, 


dA  = 


(i) 


FIG.  29. 


for  the  differential  of  the  area,  was  obtained  upon  the  supposi- 
tion that  A  and  B  were  on  the  same  side  of  C.     Then  suppos- 
ing Pz  >  Pi,  and  that  the  line  rotates   in  the  positive  direction 
as  in  figure  29,  the  differential  of  the  area  is 
positive;  and  we  notice  that  every  point  in  the 
area  generated   is  swept    over  by  the  line 
ABy   the  left  hand  side  as  we  face   in  the 
direction  A  B  preceding. 

1  74-.  We  shall  now  show  that  in   every 
case,    the    formula   requires    that    an    area 
swept  over  with    the  left  side  preceding,  shall  be  considered 
as  positively,  generated,  and    one  swept  over  in  the  opposite 
direction  as  negatively  generated. 

In  the  first  place,  if  C  is  between  A  and 
B  so  that  P!  is  negative,  as  in   figure  30,  p\ 
is  still  positive,  and  formula  (i)  still  gives 
the  difference  between  the  areas  generated 
by   CB  and   AC.      Hence  the  latter   area, 
which  is  now  generated  by  a  part   of   the 
line   AB,   must   be    regarded    as  generated 
negatively,  but  the  right  hand  side  as  we 
face  in   the   direction  AB  of  this  portion   of  the  line  is  now 
preceding,  which  agrees  with  the  rule  given  in  Art.  173. 

Again,  if  C  is  beyond  B,  the  formula  gives  the  difference 
of  the  generated  areas  ;  but  since  p?  is  numerically  greater 
than  p£,  in  this  case,  dA  is  negative,  and  the  area  generated  by 
AB  is  the  difference  of  the  areas,  and  is  negative  by  the  rule. 


f  IG.    30. 


210 


GEOMETRICAL   APPLICATIONS. 


[Art.  174- 


Finally,  if  the  direction  of  rotation  be  reversed,  d<f>  and 
therefore  dA  change  sign,  but  the  opposite  side  of  each  por- 
tion of  the  line  becomes  in  this  case  the  preceding  side. 

175.  We  may  now  put  the  expression  for  the  area  in  another 
form.  For 


whatever  be  the  signs  of  p2  and  plt  the  first  factor  is  the  length 
of  AB,  which  we  shall  denote  by  /,  and  the  second  factor  is 
the  distance  of  the  middle  point  of  AB  from  the  centre  of 
rotation,  which  we  shall  denote  by  pm.  Hence,  putting 


Pi  =  /, 


and 


+ 


we  have 


=  I  lpm  d<j>. 


(2) 


Since  pm  d$  is  the  differential  of  the  motion  of  the  middle  point 
in  a  direction  perpendicular  to  AB,  this  expression  shows  that 
the  differential  of  the  area  is  the  product  of  this  differential  by 
the  length  of  the  generating  line. 

Areas  generated   by  Lines  whose   Extremities   describe 
Closed   Circuits. 

176.  Let  us  now  suppose  the  generating  line  AB  to  move 
from  a  given  position,  and  to  return  to  the 
same  position,  each  of  the  extremities  A  and 
B  describing  a  closed  curve  in  the  positive 
direction,  as  indicated  by  the  arrows  in  figure 
31.  It  is  readily  seen  that  every  point  which 
is  in  the  area  described  by  B,  and  not  in  that 
described  by  A,  will  be  swept  over  at  least 
once  by  the  line  AB,  the  left  side  preceding, 
FIG.  31.  and  if  passed  over  more  than  once,  there  will  be 


§  XIV.]      AREAS   GENERATED  BY  MOVING  LINES.  211 

an  excess  of  one  passage,  the  left  side  preceding.  Therefore 
the  area  within  the  curve  described  by  B,  and  not  within  that 
described  by  A,  will  be  generated  positively.  In  like  manner 
the  area  within  the  curve  described  by  A,  and  not  within  that 
described  by  B,  will  be  generated  negatively.  Furthermore,  all 
points  within  both  or  neither  of  these  curves  are  passed  over, 
if  at  all,  an  equal  number  of  times  in  each  direction,  so  that  the 
area  common  to  the  two  curves  and  exterior  to  both  disap- 
pears from  the  expression  for  the  area  generated  by  AB. 

Hence  it  follows  that,  regarding  a  closed  area  whose  perimeter 
is  described  in  the  positive  direction  as  positive,  the  area  generated 
by  a  line  returning  to  its  original  position  is  the  difference  of  the 
areas  described  by  its  extremities.  This  theorem  is  evidently 
true  generally,  if  areas  described  in  the  opposite  direction  are 
regarded  as  negative. 


Amslers  Planimeter. 

177.  The  theorem  established  in  the  preceding  article  may 
be  used  to  demonstrate  the  correctness  of  the  method  by 
which  an  area  is  measured  by  means  of  the  Polar  Planimeter, 
invented  by  Professor  Amsler,  of  Schaffhausen. 

This  instrument  consists  of  two  bars,  OA  and  AB,  Fig.  32, 
jointed  together  at  A.     The  rod  OA    turns  on 
a  fixed  pivot  at  O,  while  a  tracer  at  B  is  carried      s       """""^ — v 
in   the  positive    direction    completely   around     I  \ 

the  perimeter  of  the  area  to  be  measured.     At     ^ 

some  point  C  of  the  bar  AB  a  small  wheel  is  c     

fixed,  having  its  axis  parallel  to  AB,  and  its 
circumference  resting  upon  the  paper.  When 
B  is  moved,  this  wheel  has  a  sliding  and  a  roll- 
ing motion  ;  the  latter  motion  is  recorded  by 
an  attachment  by  means  of  which  the  number  FIG.  32. 

of  turns  and  parts  of  a  turn  of  the  wheel  are  registered. 


212  GEOMETRICAL   APPLICATIONS.  [Art.  1/8. 

178.   Let  M  be  the  middle  point  of  AB,  and  let 
OA=a,  AB  =  b,  MC '  =  c. 


Since  b  is  constant,  the  area  described  by  AB  is  by  equation  (2), 
Art.  175, 

Krez.AB=b  \pmd<j) (i) 


Denoting  the  linear  distance  registered  on  the  circumference 
of  the  wheel  by  s,  ds  is  the  differential  of  the  motion  of  the 
point  C,  in  a  direction  perpendicular  to  AB,  and  since  the  dis- 
tance of  this  point  from  the  centre  of  rotation  is  pm  +  c, 

ds  =  (pm  +  c)  d(j>  : 
substituting  in  (i)  the  value    of  pmd<}>, 

)  ......     (2) 


179.  Two  cases  arise  in  the  use  of  the  instrument.  When, 
as  represented  in  Fig.  32,  O  is  outside  the  area  to  be  meas- 
ured, the  point  A  describes  no  area,  and  by  the  theorem  of 
Art.  176,  equation  (2)  represents  simply  the  area  described 

by  B.     In  this  case  </>  returns  to  its  original  value,   hence    d$ 

vanishes,  and  denoting  the  area  to  be  measured  by  A,  equation 
(2)  becomes 

A=bs  ......    :    .     .      (3) 

In  the  second  case,  when  O  is  within   the  curve  traced  by  B, 
the  point  A  describes  a  circle  whose  area  is  no1,  and  the  limit- 


§  XIV.]  AMSLER'S  PLANIMETER.  2i3 

ing  values  of  <f>  differ  by  a  complete  revolution.  Hence  in  this 
case  equation  (2)  becomes 

A  —  7taz  —  bs  —  2nbc, 
or  A  =  bs  +  7t  (a2  -  2bc}*      .....  (4) 

In  another  form  of  the  planimeter  the  point  A  moves  in  a 
straight  line,  and  the  same  demonstration  shows  that  the  area 
is  always  equal  to  bs. 

Examples  XIV. 

i.  The  involute  of  a  circle  whose  radius  is  a  is  drawn,  and  a  tangent 
is  drawn  at  the  opposite  end  of  the  diameter  which  passes  through  the 
cusp  ;  find  the  area  between  the  tangent  and  the  involute. 

a*7t  (3  +  7T>) 


2.  Two  radii  vectores  of  a  closed  oval  are  drawn  from  a  fixed  point 
within,  one  of  which  is  parallel  to  the  tangent  at  the  extremity  of  the 
other  ;  if  the  parallelogram  be  completed,  the  area  of  the  locus  of  its 
vertex  is  double  the  area  of  the  given  oval. 

3.  Show  that  the  area  of  the  locus  of  the  middle  point  of  the  chord 
joining  the  extremities  of  the  radii  vectores  in  Ex.  2,  is  one  half  the 
area  of  the  given  oval. 


*The  planimeter  is  usually  so  constructed  that  the  positive  direction  of  rotation 
is  with  the  hands  of  a  watch.  The  bar  b  is  adjustable,  but  the  distance  A  C  is  fixed 
so  that  c  varies  with  b.  Denoting  A  C  by  q,  we  have  c  =  q  —  \b,  and  the  constant 
to  be  added  becomes  C=  it  (a3  —  zbq  +  32)  in  which  a  and  q  are  fixed  and  b  adjusta- 
ble. In  some  instruments  q  is  negative. 

It  is  to  be  noticed  that  in  the  second  case  s  may  be  negative  ;  the  area  is  then 
the  numerical  difference  between  the  constant  and  bs. 


214  GEOMETRICAL   APPLICATIONS.  [Ex.  XIV. 

4.  Prove  that  the  difference  of  the  perimeters  of  two  parallel  ovals, 
whose  distance  is  b,  is  27tb,  and  that  the  difference  of  their  areas  is 
the  product  of  b  and  the  half  sum  of  their  perimeters. 

5.  From  a  fixed  point  on  the  circumference  of  a  circle  whose  radius 
is  a  a  radius  vector  is  drawn,  and  a  distance  b  is  measured  from  the 
circumference  upon  the  radius  vector  produced  ;  the  extremity  of  b 
therefore  describes  a  limacon  :  show  that  the  area  generated  by  b  when 
b  >  20,  is  the  area  of  the  limacon  diminished  by  twice  the  area  of  the 
circle,  and  thence  determine  the  area  of  the  limacon. 

71(20"  +  £'). 

6.  Verify  equation  (4),  Art.  179,  when  the  tracer  describes  the 
circle  whose  radius  is  a  +  b. 

7.  Verify  the  value  of  the  constant  in  equation  (4),  Art.    179,  by 
determining  the  circle  which  may  be  described  by  the  tracer  without 
motion  of  the  wheel. 

8.  If,  in  the  motion  of  a  crank  and  connecting  rod  (the  line  of  motion 
of  the  piston  passing  through  the  centre  of  the  crank),  Amsler's  record- 
ing wheel  be  attached  to  the  connecting  rod  at  the  piston  end,  deter- 
mine s  geometrically,  and  verify  by  means  of  the  area  described  by  the 
other  end  of  the  rod. 

9.  The  length  of  the  crank  in  Ex.  8  being  a,  and  that  of  the  con- 
necting rod  b,  find  the  area  of  the  locus  of  a  point  on  the  connecting 
rod  at  a  distance  c  from  the  piston  end. 


10.  If  a  line  AB  of  fixed  length  move  in  a  plane,  returning  to  its 
original  position  without  making  a  complete  revolution,  denoting  the  areas 
of  the  curves  described  by  its  extremities  by  (A)  and  (B},  determine 
the  area  of  the  curve  described  by  a  point  cutting  AB  in  the  ratio 
m  :  n. 

n(A)  +  m(JB) 
m  +  » 


§  XIV.] 


EXAMPLES. 


215 


ii.  If  the  line  in  Ex.  10  return  to  its  original  position  after  making  a 
complete  revolution,  prove  Holditch's  Theorem  ;  namely,  that  tne  area  of 
the  curve  described  by  a  point  at  the  distance  c  and  c  from  A  and  B  is 


c'(A] 


, 


12.  Show  by  means  of  Ex.  n  that,  if  a  chord  of  fixed  length  move 
around  an  oval,  and  a  curve  be  described  by  a  point  at  the  distances 
c  and  c  from  its  ends,  the  area  between  the  curves  will  be  ncc  '. 


XV. 

Approximate  Expressions  for  Areas  and  Volumes. 

180.  When  the  equation  of  a  curve  is  unknown,  the  area 
between  the  curve,  the  axis  of  x,  and 
two  ordinates  may  be  approximately  ex- 
pressed in  terms  of  the  base  and  a  lim- 
ited number  of  ordinates,  which  are  sup- 
posed to  have  been  measured. 

Let  ABCDE  be  the  area  to  be  de- 
termined ;  denote  the  length  of  the  base 
by  2h ;  and  let  the  ordinates  at  the  ex- 
tremities and  middle  point  of  the  base 
be  measured  and  denoted  by  y^y*  andjys.  Taking  the  base  for 
the  axis  of  x,  and  the  middle  point  as  origin,  let  it  be  assumed 
that  the  curve  has  an  equation  of  the  form 

(i) 


then  the  area  required  is 

f*  BX*      Cx*      Dx^~\h 

A=\     ydx-Ax-\- —  H +-  = 

J-A  2          3          4  J_A 

in  which  which  A  and  C  are  unknown. 


,. 
,    .(2) 


2l6 


GEOMETRICAL   APPLICATIONS 


[Art.   1 80. 


In   order  to  express   the    area  in   terms   of  the  measured 
ordinates,  we  have  from  equation  (i), 

jt  =  A  +  Bh  +  Ch*  +  Dh\ 


whence  we  derive 


y\ 

472 


and  substituting  in  (2), 


It  will  be  noticed  that  this  formula  gives  a  perfectly  ac 
curate  result  when  the  curve  is  really  a  parabolic  curve  of  the 
third  or  a  lower  degree. 

181.  If  the  base  be  divided  into  three  equal  intervals,  each 
denoted  by  h,  and  the  ordinates  at  the  extremities  and  at  the 
points  of  division  measured,  we  have,  by  assuming  the  same 
equation, 


(i) 


From  the  equation  of  the  curve, 


.x 

>       ?                         2 

£/* 

"4               8 

^7^2      2)ffi 

>H~ 

->2~                 2 
-  ^         ^ 

T"  ^-' 

a2    OT 

.74         ^    1-    * 
FIG.  -u..                                                      2 

+     4     +"8       J 

§  XV.]  SIMPSON'S  RULES.  2 1/ 


whence  }\  +  y±  —  2A  +  -     — , 

Ch* 

2 


From  these  equations  we  obtain 


A 


9/3-74 


6 
and  €»=*- 


4 
Substituting  in  equation  (i), 


Simpsons  Rules. 

182.  The  formulae  derived  in  Articles  iSoand  181,  although 
they  were  first  given  by  Cotes  and  Newton,  are  usually  known 
as  Simpson  's  Rules,  the  following  extensions  of  the  formulae 
having  been  published  in  1743,  in  his  Mathematical  Disserta- 
tions. 

If  the  whole  base  be  divided  into  an  even  number  n  of 
parts,  each  equal  to  //,  and  the  ordinates  at  the  points  of  divis- 
ion be  numbered  in  order  from  end  to  end,  then  by  applying 
the  first  formula  to  the  areas  between  the  alternate  ordinates, 
we  have 


That  is  to  say,  the  area  is  equal  to  the  product  of  the  sum  of 
the  extreme  ordinates,  four  times  the  sum  of  the  even-num- 


21 8  GEOMETRICAL   APPLICATIONS.  [Art.   182 

bered  ordinates,  and  twice  the  sum  of  the  remaining  odd-num- 
bered ordinates,  multiplied  by  one  third  of  the  common  interval. 
Again,  if  the  base  be  divided  into  a  number  of  parts  divis- 
ible by  three,  we  have,  by  applying  the  formula  derived  in 
Art.  181  to  the  areas  between  the  ordinates  y± y^y^y^  and  so  on, 

\h 

^  =       ( Ji  +  3J2  +  373  +  2j4  +  3j5  •  •  •  +  $yn 


Cotes    Method  of  Approximation. 

183.  The  method  employed  in  Articles  180  and  181  is 
known  as  Cotes  Method.  It  consists  in  assuming  the  given 
curve  to  be  a  parabolic  curve  of  the  highest  order  which  can 
be  made  to  pass  through  the  extremities  of  a  series  of  equi- 
distant measured  ordinates. 

The  equation  of  the  parabolic  curve  of  the  «th  order  con- 
tains n  +  i  unknown  constants;  hence,  in  order  to  eliminate 
these  constants  from  the  expression  for  an  area  defined  by  the 
curve,  it  is  in  general  necessary  to  have  n  +  I  equations  con- 
necting them  with  the  measured  ordinates.  Hence,  if  n  de- 
note the  number  of  intervals  between  measured  ordinates  over 
which  the  curve  extends,  the  curve  will  in  general  be  of  the 
degree.* 


*  If  H  denotes  the  whole  base,  the  first  factor  is  always  equivalent  to  H 
divided  by  the  sum  of  the  coefficients  of  the  ordinates  ;  for  if  all  the  ordinates  are 
made  equal,  the  expression  must  reduce  to  Hy±.  Thus,  each  of  the  rules  for  an 
approximate  area,  including  those  derived  by  repeated  applications,  as  in  Art.  182, 
may  be  regarded  as  giving  an  expression  for  the  mean  ordinate.  The  coefficients 
of  the  ordinates,  according  to  Cotes'  method,  for  all  values  of  n  up  to  n  =  IO,  may 
be  found  in  Bertrand's  Cakul  Integral,  pages  333  and  334.  For  example  (using 
detached  coefficients  for  brevity),  we  have,  when  n  —  4, 

H  r 
A  =—\.l'  32,  I2>  32,  7J; 

and  when  n  =  6, 

TT 

A  —  g  —  [41,  216,  27,  272,  27,  216,  41], 


§  XV.]  THE  FIVE-EIGHT  RULE.  219 

I84-.  For  example,  let  it  be  required  to  determine  the  area 
between  the  ordinates  y^  and  yz,  in  terms  of  the  three  equi- 
distant ordinates  y^  y»  and  y.6t  the  common  interval  being  k. 
We  must  assume 

y  -  A  +  Bx  + 


then  taking  the  origin  at  the  foot  of  yl} 

(h  ,  T  /      Bh 

A  =\y dx  —  h\   A  +  —  + 

from  which  A,  B  and  C  must  be   eliminated  by  means  of  the 
equations 

yz  =  A+Bh  +  Ch\ 

y*  —  A  - 

Solving  these  equations,  we  obtain 

A  =  n, 


If  we  make  a  slight  modification  in  the  ratios  of  these  last  coefficients  by  sub- 
stituting for  each  the  nearest  multiple  of  42,  we  have 

A  =  - —  [42,  210,  42,  252,  42,  210,  42], 
840 

(the  denominator  remaining  unchanged,  since  the  sum  of  the  coefficients  is  still 
840),  which  reduces  to 

^  =  —[1,5,  1,6,  1,5,  ij- 

This  result  is  known  as  Weddles  Rule  for  six  intervals.  The  value  thus  given  to 
the  mean  ordinate  is  evidently  a  very  close  approximation  to  that  resulting  from 
Cotes'  method,  the  difference  being 


220  GEOMETRICAL   APPLICATIONS.  [Art.  184. 

and  substituting 


1  — 

185.  It  is,  however,  to  be  noticed,  that  when  the  ordinates 
are  symmetrically  situated  with    respect  to  the  area,  if  n  is 
even,  the   parabolic  curve  may  be  assumed  of   the   (n  +  i)tli 
degree.     For   example,   in  Art.  180,  n  =  2,  but  the   curve  was 
assumed   of  the    third   degree.     Inasmuch    as  A,  B,  C  and  D 
cannot  all  be  expressed  in  terms  of  }\,  y.^  and  yz,  we  see  that  a 
variety  of  parabolic  curves  of  the  third  degree  can  be  passed 
through  the  extremities  of  the  measured  ordinates,  but  all  of 
these  curves  have  the  same  area.* 

Application  to  Solids. 

186.  If  y  denotes  the  area  of  the  section  of  a  solid  perpen- 
dicular to  the  axis  of  x,  the  volume  of  the  solid  is    y  dx,  and 

*  This  circumstance  indicates  a  probable  advantage  in  making  n  an  even  num- 
ber when  repeated  applications  of  the  rules  are  made.  Thus,  in  the  case  of  six 
intervals,  we  can  make  three  applications  of  Simpson's  first  rule,  giving 

TT 

A~^  t1-  4,  2,  4,  2,  4,  i]  ........     (i> 

or  two  of  Simpson's  second  rule,  giving 

A  -  —^  [i,  3,  3,  2,  3,  3,  i]  ........     (2> 

In  the  first  case,  we  assume  the  curve  to  consist  of  three  arcs  of  the  third  degree, 
meeting  at  the  extremities  of  the  ordinates  y3  and/5  ;  but,  since  each  of  these  arcs 
contains  an  undetermined  constant,  we  can  assume  them  to  have  common  tangents 
at  the  points  of  meeting.  We  have  therefore  a  smooth,  though  not  a  continuous 
curve.  In  the  second  case,  we  have  two  arcs  of  the  third  degree  containing  no 
arbitrary  constants,  and  therefore  making  an  angle  at  the  extremity  of  _j/4.  It  is 
probable,  therefore,  that  the  smooth  curve  of  the  first  case  will  in  most  cases  form  a 
better  approximation  than  the  broken  curve  of  the  second  case. 

In  confirmation  of  this  conclusion,  it  will  be  noticed  that  the  ratios  of  the 
coefficients  in  equation  (i)  are  nearer  to  those  of  Cotes'  coefficients  for  n  =  6,  given 
in  the  preceding  foot-note,  than  are  those  in  equation  (2). 


§XV.] 


APPLICATION    TO   SOLIDS. 


221 


therefore  the  approximate  rules  deduced  in  the  preceding  arti- 
cles apply  to  solids  as  well  as  to  areas.  Indeed,  they  may  be 
applied  to  the  approximate  computation  of  any  integral,  by 
putting  y  equal  to  the  coefficient  of  dx  under  the  integral  sign. 
The  areas  of  the  sections  may  of  course  be  computed  by 
the  approximate  rules. 


Woollens  Rule. 

187.  When  the  base  of  the  solid  is  rectangular,  and  the 
ordinates  of  the  sections  necessary  to  the  application  of  Simp- 
son's first  rule  are  measured,  we  may,  instead  of  applying  that 
rule,  introduce  the  ordinates  directly  into  the  expression  for 
the  area  in  the  following  manner. 

Taking  the  plane  of  the  base  for  the  plane  of  xy,  and  its 
centre  for  the  origin,  let  the  equation  of  the  upper  surface  be 
assumed  of  the  form 


Let  2,h  and  2k  be  the  dimensions  of  the  base,  and  denote 
the  measured  values  of  z  as  indicated  in 
Fig.  35«     The  required  volume  is  |Z 


th    tk 
V-\          zdydx. 

}  -h  )  -k    ' 


This  double  integral  vanishes  for  every 
term  containing  an  odd  power  of  x  or  an 
odd  power  of_^:  hence 


— 


222  GEOMETRICAL  APPLICATIONS.  [Art.  iS/. 

By  substituting  the  values  of  x  and  y  in  the  equation  of  the 
surface,  we  readily  obtain 


(2) 

,     ...    (3) 
.  ...    (4) 

From  these  equations  two  very  simple  expressions  for  the 
volume  may  be  derived  ;  for,  employing  (2)  and  (4),  equation 
(i)  becomes 

ihk 
r==j?(a>  +  &i  +  2h  +  &9  +  cd;    .    .    .    .  (5) 

and  employing  (2)  and  (3), 

hk 
V-  —  fa  +  (h  +  8^2  +  ^  +  c3)  .....    (6) 

o 

Equation  (5)  is  known  as  Woolleys  Rule;  the  ordinates  employed 
are  those  at  the  middles  of  the  sides  and  at  the  centre  ;  in  (6), 
they  are  at  the  corners  and  at  the  centre. 


Examples  XV. 

1.  Apply  Simpson's  Rule  to  the  sphere,  the  hemisphere,  and  the 
cone,  and  explain  why  the  results  are  perfectly  accurate. 

2.  Apply  Simpson's  Second  Rule  to  the  larger  segment  of  a  sphere 
made  by  a  plane  bisecting  at  right  angles  a  radius  of  the  sphere. 


§  XV.]  EXAMPLES.  223 

3.  Find  by  Simpson's  Rule  the  volume  of  a  segment  of  a  sphere, 
b  and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 

f  W  +  V>  +  V). 

4.  Find  by  Simpson's  Rule  the  volume  of  the  frustum  of  a  cone, 
b  and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 


5.   Compute  by  Simpson's  First  and  Second  Rules  the  value  of 

dx 

,  the  common  interval  being  T^  in  each  case. 


J  o  I    +  X 

The  first  rule  gives  0.69314866;  the  second  rule  gives  0.69315046. 
The  correct  value  is  obviously  loge  2  =  0.69314718.* 

6.  Find  the  volume  considered  in  Art.  187,  directly  by  Simpson's 
Rule,  and  show  that  the  result  is  consistent  with  equations  (5)  and  (6). 


7.  Find,  by  elimination,  from  equations  (5)  and  (6),  Art.  187,  a 
formula  which  can  be  used  when  the  centre  ordinate  is  unknown. 

V  =  —  [4(af  +£,  +  £,+  £t)  -  (a,  +  a,  +  c}  +  Ol- 

O 

*  The  error  in  the  eighth  place  of  decimals  is  therefore  148  by  the  First 
Rule,  and  328  by  the  Second  Rule,  the  First  Rule  giving  the  better  result  as 
anticipated  in  the  foot-note  of  p.  220.  Using  the  same  ordinate?,  Weddles' 
Rule  (see  foot-note,  p.  219)  gives  the  extremely  accurate  result  0.69314722,  the 
error  being  only  4  in  the  eighth  place. 


224  MEAN  VALUES  AND    PROBABILITIES.      [Art.  1 88. 

CHAPTER  IV. 

MEAN  VALUES  AND  PROBABILITIES. 


XVI. 
The  Average  or  Arithmetical  Mean. 

188.  A  MEAN  of  several  values  of  a  quantity  is  an  interme- 
diate value  such  that,  when  it  is  substituted  for  each  of  the 
given  values  in  performing  a  certain  operation,  the  result  is 
unchanged.  For  example,  if  the  result  of  the  operation  is  the 
product  of  the  given  values,  the  mean  value  found  is  that 
known  as  the  geometrical  mean.  But  the  usual  and  most 
simple  mean  value  is  that  obtained  when  the  operation  is  that 
of  summation.  This  is  known  as  the  average  or  arithmetical 
mean*  Thus,  if  M  denotes  the  average  of  n  values  of  y  de- 
noted by  y^  ,  yz  ,  .  .  .  yn  ,  we  have 


(l) 


Here  the  aggregate  represented  by  either  member  of  equa- 
tion (i)  is  the  same  whether  the  n  given  values  of  y  be  taken 
or  the  mean  value  be  taken  n  times. 


*  A  mean  value  of  any  other  kind  can  usually  be  defined  by  the  aid  of  the 
arithmetical  mean.  Thus,  the  geometrical  mean  is  the  quantity  whose  logarithm 
is  the  mean  of  those  of  the  given  quantities ;  the  harmonic  mean  is  that  whose 
reciprocal  is  the  mean  of  the  reciprocals.  The  mean  error  in  the  Theory  of  Least 
Squares  is  the  error  whose  square  is  the  mean  of  the  squares  of  the  errors. 


§  XVI.]    THE   MEAN  OF  A    CONTINUOUS    VARIABLE.          22$ 

189.  When  a  number  pl  of  the  quantities  has  a  common 
value  jj/j,  a  number  pz  has  a  common  value  yz,  and  so  on,  the 
total  number  n  of  formula  (V  is  equal  to  pl  -\-  p2  -f-  .  .  .  or  2p, 
and  the  formula  becomes 

^p.M=  /^i  +  A^2  +..«==  ^/J-       •      .     (2) 

The  numbers  A»  A>  •  •  •  are  ca^e<^  the  weights  of  jj/j,  jj/2, 
.  .  .,  and  the  mean  is  called  the  weighted  arithmetical  mean 
of  the  several  values. 

The  mean  value  of  a  quantity  which  admits  of  a  continuous 
series  of  values  (which  is  the  subject  of  the  present  Chapter)  is 
a  modification  of  the  M  of  equations  (i)  and  (2),  in  deriving 
which,  integration  takes  the  place  of  summation. 


The  Mean  of  a  Continuous  Variable. 

190.  Consider  all  the  values  of  a  variable  which  varies  con- 
tinuously between  certain  extreme  values,  and  suppose  a  large 
number  n  of  these  values  to  be  chosen  and  their  mean  taken. 
Then,  supposing  the  manner  of  selection  to  be  such  that  we 
can  pass  to  the  limit  when  n  is  indefinitely  increased,  we  shall 
have  a  mean  value  which  depends  upon  all  the  values  of  the 
variable  in  question,  and  is,  therefore,  properly  called  a  mean 
value  of  the  variable.      But  the  value  of  this  mean  obviously 
depends    also  upon  the  method  in  which  the  n  values  were 
selected.       For  example,    in  finding  the   mean   velocity  of  a 
point  describing  a  straight  line  with  variable  velocity,  we  shall 
arrive  at  a  certain  result  if  we  take  the  n  values  of  the  velocity 
at  equal  intervals  of  time  ;   but  the  result  will  be  different  if  we 
select  the  velocities  with  which  the  point  passes  equally  dis- 
tant points  of  its  path. 

191.  When  the  variable  in  question  is  a  function  of  some 
single  independent  variable,  the  mean  obtained  by  taking  equi- 


226 


MEAN    VALUES  AND    PROBABILITIES.     [Art.  191. 


distant  values  of  the  independent  variable  is  called  tlie  mean 
value  of  the  function  for  the  range  of  values  given  to  the  inde- 
pendent variable.  Let  y  =  f(x]  be  the  function,  represented, 
as  in  Fig.  8,  p.  1 2 1 ,  by  the  curve  CD ;  then  the  figure  illustrates 
the  mode  in  which  n  values  of  the  function  or  ordinate  are 
taken  in  finding  the  mean  value  of  the  function  for  all  values 
of  x  between  OA  =  a  and  OB  =  b.  These  values  are  the 
y\ »  }>i  >  i  •  *  y*  °f  Art.  99  erected  at  the  common  interval  Ax, 
where  n  Ax  =  b  —  a. 

Now,  multiplying  equation  (i),  Art.  188,  by  Ax,  we  have 

n  Ax-M  =  ylAx  -f-  y2  Ax  -f  .  .  .  ~\-ynAx, 
or 

(b  _  a)M  =  ^yAx. 

Here  M  is  the  mean  of  the  n  actual  values  of  y,  and  the 
mean  value  required  is  found  by  passing  to  the  limit  when  n  is 
indefinitely  increased. 

Ji 
y  dx, 
a 

which  is  the  area  CABD  in  Fig.  8.  Hence  the  mean  value 
of ./(•*•)  for  all  values  of  x  between  a  and  b  is  given  by  the 
equation 


(3) 


192.  The  expression  mean  ordinate  of  any  portion  of  a 
curve  is  always,  unless  otherwise  stated,  held  to  signify  the 
mean  ordinate  regarded  as  a  function  of  the  abscissa.  Hence 
it  is  the  height  of  the  rectangle  of  which 
the  base  is  the  projection  of  the  curve  on 
the  axis  of  x,  and  the  area  is  equal  to 
that  included  between  the  curve,  the 
base  and  the  extreme  ordinates.  Thus, 
"the  mean  ordinate"  of  a  semicircle 

z2,  by  the 


\ 


R 


O 

FIG.  36. 
whose  radius  is  a  is  found  by  dividing  the  area, 


§  XVI.]      MEAN  OF  EQUALLY  PROBABLE    VALUES.  22/ 

base  2a\  whence  M  =  \na.  This  is  the  average  value  of  per- 
pendiculars erected  at  equal  distances  along  the  diameter.  See 
Fig.  36. 

193.  On  the  other  hand,  if  the  average  value  of  perpendic- 
ulars let  fall  from  equidistant  points  on 
the  arc   is  required   (see   Fig.    37),  it  is 
necessary  to  express  the  perpendicular 
as  a  function   of  the  arc  or  angle  sub-    / 


tended  at  the  centre.      Denoting  this  (as  ° 

measured  from  one  extreme  radius)  by  FlG-  37 

6,  the  perpendicular  is  a  sin  0,  and  the  value  of  this  mean  is, 
by  equation  (3), 

e* 
a\  sin  6  d9 

M= 


71 


The  Mean  of  Equally  Probable  Values. 

1 94-.  The  expression  mean  value  of  a  variable  quantity 
selected  under  given  circumstances  is  often  used  to  designate 
the  mean  of  all  the  values  which  are  equally  probable  under 
the  circumstances.  A  point  is  said  to  be  taken  at  random 
upon  a  line  of  given  length  when  it  is  equally  likely  to  fall 
upon  any  one  of  any  equal  segments  of  the  line.  Hence  the 
first  of  the  mean  values  of  PR  found  in  the  preceding  articles 
may  be  called  the  mean  value  when  R  is  taken  at  random 
upon  the  diameter,  and  the  second  is  the  mean  value  when  P 
is  taken  at  random  upon  the  semicircumference. 

So  also  M  in  equation  (i),  Art.  188,  is  the  mean  value  of 
y  whenj^,  yz,  .  .  .  yn  are  equally  probable,  and  are  the  only 
possible  values  of  y.  In  this  case,  the  finite  number  n  is  the 
total  number  of  cases  which  are  possible  and  equally  probable. 

It  will  be  convenient  also  in  equation  (3),  Art.  191,  to  speak 


228  MEAN    VALUES  AND   PROBABILITIES.    [Art.   194. 

of  b  —  a  (which  takes  the  place  of  n  when  it  is  increased  in- 
definitely in  passing  to  the  limit)  as  the  total  number  of  cases, 
and  of  the  definite  integral  in  the  second  member  as  the  aggre- 
gate of  them's  in  the  total  number  of  cases. 


The  Mean  of  a  Function  of  Two^  Variables. 

195.  When  the  quantity  whose  mean  is  required  depends 
"upon  two  variables  admitting  of  continuous  values  under  cer- 
tain restrictions,  we  may  represent  it  by  the  ^-coordinate  of  a 
surface  of  which  the  independent  variables  x  and  y  are  the  other 
two  rectangular  coordinates.      The  restrictions  imposed  upon 
the  values  of  x  and  y  now  limit  the  foot  of  the  ^-coordinate  to 
a  certain  area  in  the  plane  of  xy.      If  these  restrictions  consist 
of  fixed  limits  between  which  x  must  lie,  together  with  other 
fixed  limits  between  which  y  must  lie,  this  area  is  a  rectangle. 
It  may,  however,  be  an  area  of  any  other  shape;   the  limiting 
values  of  y  will  then  be  different  for  different  values  of  x.      In 
•choosing  the  points  for  which  the  values  of  z  are  taken  in  form- 
ing the  mean,  the  values  of  y,  for  a  given  value  of  x,  are  sup- 
posed to  be  taken  at  equal  intervals  Ay  between  the  limits 
.corresponding  to  that  particular  value  of  x.      In  like  manner, 

the  values  of  x  chosen  are  taken  at  equal  intervals  Ax,  so  that 
the  points  (x,  y]  are  uniformly  distributed  over  the  area. 
Thus,  if  the  area  contains  a  large  number,  n,  of  elementary 
rectangles  of  dimensions  Ay  and  Ax,  then  one  value  of  z  is 
taken  corresponding  to  each  element,  Ay  Ax,  of  area. 

196.  Now,  putting  z  for  y  in  equation  (i),  Art.  188,  and 
multiplying  each  member  by  Ay  Ax,  we  have 

n  Ay  Ax-M •=  "2 zAyAx (l) 

On  passing  to  the  limit  when  Ax  and  Ay  are  indefinitely 


XVI.]  MEAN  OF  A  FUNCTION  OF  TWO   VARIABLES.        229 


decreased,  M becomes  the  mean  value  required.      The  limiting- 
value  of  nAyAx  is  the  area  mentioned  above;  and  its  value  is 


=   \\dydx, 


where  the  limits  of  integration  are  given  in  the  form  of  restric- 
tions upon  the  values  of  x  and  y.  The  limiting  value  of  the 
second  member  of  equation  (i)  is  the  double  integral 


\zdydx, 


taken  with  the  same  limits;  or,  as  expressed  in  Art.  126,  in- 
tegrated over  the  area  A;  and  its  value,  as  shown  in  that 
article,  is  the  volume  of  the  cylindrical  solid  whose  upper  sur- 
face is  in  the  surface  2  =p  <p(x,y),  and  whose  base  is  the  area 
A.  Thus  we  have  the  equation 

M-  \  \dy  dx  =  \\  <P(x,  y}dy  dx,         ...      (2) 

which  defines  the  mean  value  of  a  function  of  two  variables, 
and  shows  that,  when  thus^geometrically  represented,  it  is  the 
height  of  a  cylinder  having  the  area  of  integration  as  base  and 
a  volume  equal  to  that  of  the  solid  described  above. 

(97.  The  "  mean  ordinate  "  for  a  given  surface  referred  to 
rectangular  coordinate  planes  is  always  understood  to  signify 
the  mean  value  of  z,  thus  considered  as  a  function  of  x  and  y. 
If  the  volume  and  the  area  of  the  base  are  known,  the  mean 
is  at  once  known.  For  example,  the  mean  ordinate  of  a  hemi- 
spherical surface  whose  radius  is  a  (of  which  Fig.  36  may  rep- 
resent a  central  section)  is  given  by 

7iaz-M=     7ra3,          whence         M=    a. 


23O  MEAN    VALUES   AND  PROBABILITIES.     [Art.    197. 

When  regarded  as  the  mean  of  equally  probable  values, 
as  in  Art.  194,  it  is  implied  that  every  admissible  pair  of  values' 
of  x  and  y  has  the  same  probability.  That  is  to  say,  in  the 
geometrical  illustration,  every  position  of  the  point  R  within 
the  area  is  equally  probable,  and  this  is  expressed  by  saying 
that  the  point  R  falls  at  random  upon  the  base. 

It  will  be  convenient  here  also  (compare  Art.  194)  to 
speak  of  the  area,  which  in  equation  (2)  of  Art.  196  takes  the 
place  of  n,  as  tJie  total  number  of  equally  probable  cases;  and 
of  the  second  member  of  the  equation  as  the  aggregate  of  the 
function  z  in  the  total  number  of  cases. 

The  Mean  of  Values  not  Equally  Probable. 

198.  When  the  variable  in  question  is  expressed  as  a  func- 
tion of  a  single  variable,  it  may  happen  that  the  mean  required 
is  not  that  which  we  have  defined  as  the  mean  value  of  tJie 
function  between  certain  limits.  For  example,  if  the  average 
height  above  the  base  of  a  point  on  the  surface  of  the  hemi- 
sphere is  required,  it  may  be  expressed  by  a  sin  6,  as  in  Fig. 
37  (which  we  now  take  to  represent  a  central  section  perpen- 
dicular to  the  circular  base).  But  the  mean  now  required  is 
not  the  mean  value  of  PR  considered  as  a  function  of  0  (which 
was  found  in  Art.  193) ;  for  all  values  of  B  are  not  now  equally 
probable.  In  the  present  problem,  the  "total  number  of 
cases  ' '  is  represented  by  the  area  of  the  convex  surface  of  the 
hemisphere  upon  which  P  is  said  to  fall  at  random.  Instead 
of  a  single  case,  or  an  equal  number  of  cases,  corresponding 
to  each  element  of  arc  add,  we  have  a  number  proportional  to 
the  area  of  the  element  of  surface  dS  which  corresponds  to 
that  element  of  arc,  and  the  surface  elements  corresponding 
to  different  elements  of  arc  are  unequal  areas.  Thus  we  have 
a  case  of  the  weighted  mean  of  Art.  189;  in  which  the  weight 
p  is  represented  by  the  element  of  surface,  and  ~2p  by  the 


§  XVI.]  MEAN   OF    VALUES  NOT  EQUALLY  PROBABLE.    2^1 

whole  surface.  The  length  of  the  element  dS  is  the  circum- 
ference whose  radius  is  a  cos  6;  hence 

dS  —  2710*  cos  6dO, 

and  the  surface  of  the  hemisphere  is  2710*.  Thus  M  is  deter- 
mined by 

jr    ' 

27ta*  •  M  =   [a  sin  6  dS  =  2  no?  f2  sin  0  cos  0  dO  =  7tas, 

which  gives  M  —  \a  for  the  average  height  of  a  point  on  the 
surface  of  the  hemisphere. 

199.  In  the  foregoing  problem,  if  P  be  joined  to  the  centre 
O,  6  is  the  elevation  of  the  direction  OP  above  the  horizontal 
plane  (that  of  the  base  of  the  hemisphere).  OP  is  said  to  have 
a  random  direction  in  space  because  P  was  taken  at  random 
upon  the  spherical  surface.  Hence  the  problem  solved  above 
is  the  same  as  that  of  finding  the  mean  initial  vertical  velocity 
of  a  body  projected  upward  in  a  random  direction  with  the 
velocity  a. 

Again,  the  horizontal  velocity  of  such  a  body  with  initial 
velocity  V  is  V  cos  0.  Hence  the  mean  horizontal  velocity 
is  found  by 


2 

whence  M=  %7tV. 

200.  Even  when  we  use  the  ultimate  element  of  surface 
upon  which  the  variable  point  falls  at  random  (which  may  be 
denoted  by  d^S],  and  perform  a  double  integration,  the  area 
of  the  element  may  be  regarded,  as  in  Art.  198,  as  giving  the 
weight  to  be  attributed  to  that  special  value  of  the  function 
which  corresponds  to  the  element.  It  is  only  in  the  case  of 
the  plane  element  dy  dx  that  the  area  of  an  element  is  con- 
stant, so  as  to  give  equal  weights  to  values  selected  one  from 
each  element. 


232  MEAN    VALUES  AND    PROBABILITIES.     [Art.  2OO. 

For  example,  in  using  polar  coordinates,  let  the  area  in 
Fig.  22,  p.  172,  represent  the  limits  of  integration,  and  let  one 
value  of  the  quantity  whose  mean  is  required  be  selected  for 
each  of  the  small  areas  represented.  Then,  it  is  only  by  using 
the  varying  value  of  the  area  of  an  element,  namely  r  dr d&, 
that  we  give  proper  weight  to  each  value  to  represent  uniform 
distribution  over  the  area.  If  we  use  dr  dO  simply,  we  shall 
find  quite  a  different  mean  value,  having  thus  given  more 
weight  comparatively  to  values  of  the  function  corresponding 
to  small  values  of  r  than  should  be  done  to  represent  uniform 
distribution. 


The  Centre  of  Position  ofn  Points. 

201.  The  mean  distances  of  n  points  in  a  plane  from  fixed 
straight  lines  in  the  plane,  and  more  generally  of  n  points  in 
space  from  fixed  planes,  form  an  important  application  of  the 
principles  of  mean  values. 

In  the  case  of  points  in  a  plane,  let  their  positions  be  referred 
to  rectangular  axes,  of  which  that  of  y  is  the  line  from  which 
the  mean  distance  is  required.  Then,  if  the  points  are  (x^,y^), 
(xz,y^,  .  .  .  (xn,  yn),  and  ~x  denotes  the  mean  distance  from 
the  axis  of  j/,  its  value  is 


(0 


In  like  manner,  the  mean  distance  of  the  points  from  the  axis 
of  x  is 


202.  We  shall  now  show  that  the  point  which  has  these 
mean  values  for  its  coordinates  has  the  property  that  its  dis- 


§  XVI.]  CENTRE   OF  POSITION   OF  n   POINTS.  233 

tance  from  any  straight  line  in  the  plane  is  the  mean  of  the 
distances  of  the  several  points  from  that  line. 

In  the  first  place,  let  the  line  in  question  be  parallel  to  an 
axis,  say  that  of  y,  and  at  a  distance  h  from  it.  Then  (taking 
h  as  positive  when  the  line  is  on  the  left  of  the  axis),  the  dis- 
tance of  (x,  ,  y^)  from  it  is  xl  -j-  h.  In  like  manner,  the  distance 
of  (x2,  j>2)  is  xz  -f-  ky  and  so  on.  Hence  the  mean  distance  is 

—  2(x  -\-  nty  =  —  ~2x  -j-  h  =  ~x  -f-  h,  by  equation  (i).     But  this 

is  the  distance  of  the  point  (x,  y)  from  the  line  in  question. 

Next  let  the  line  be  oblique  to  the  axes.  It  is  shown  in 
Analytical  Geometry  that  the  perpendicular  upon  a  line  from 
(x,  y)  is  of  the  form 


where  A,  B,  and  C  are  certain  constants;  in  other  words,  p 
is  a  linear  function  of  x  andj.  Hence  we  have,  for  the  dis- 
tances of  (xl,y^),  (xz>y^,  e^c-»  from  the  line 

A  =  Ax,  +  By,  +  C, 
/,  =  Ax,  +  By,  +  C, 


and  adding, 

2    =  A  •  ^x  --  B.  2         nC. 


Dividing  by  n,  the  mean  distance  is,  by  equations  (i)  and  (2), 


but  this  is  the  distance   of  the   point   (x,  ~y)   from  the  given 
oblique  line. 


234  MEAN    VALUES  AND    PROBABILITIES.     [Art.  2O2. 

The  point  (J,  ~y)  whose  position  thus  determines  the  mean 
distance  of  the  given  points  from  any  straight  line  in  the  plane 
is  called  their  Centre  of  Position. 

203.  In  like  manner,  when  n  points  Pl  ,  P2,  .  .  .  Pn  in 
space  are  referred  to  rectangular  planes,  their  mean  distances 
from  these  planes  are 


and  it  can  be  shown  that  the  mean  distance  of  the  points  from 
any  plane  is  the  distance  of  the  point  (5F,  y,  ~z)  from  that  plane. 
The  point  which  thus  determines  the  mean  distance  from  all 
planes  is  called  the  Centre  of  Position  of  the  points.  It  is  also 
called,  from  its  occurrence  in  Statics,  the  Centre  of  Gravity  of 
n  equal  particles  situated  at  the  points  Plt  P2,  .  .  .  Pn.  • 

204-.  If  a  number  pl  of  the  equal  particles  coincide  at  the 
point  Pl  ,  p2  of  them  at  P2,  and  so  on,  the  centre  of  gravity  of 
the  particles  is  at  a  distance  from  any  given  plane  which  is  the 
weighted  mean  (Art.  1  89)  of  the  distances  of  the  point  Pl  ,  P2  , 
etc.  from  the  plane.  Thus,  referring  to  rectangular  coordinate 
planes,  its  distance  x  from  that  of  yz  is  given  by  the  equation 


Multiplying  both  sides  by  the  mass  of  one  of  the  equal  particles, 
the  equation  becomes 


=  mlxl  -j-  m??2  +  •  •  •  =  ^>mx,       •      •      (0 


where  Wj  ,  m2,  etc.  are  the  masses  of  unequal  particles  situated 
at  Pj,  Pv  etc.,  and  2m  is  the  total  mass. 

The  point  (!r,  ~y,  ~z)  whose  three  coordinates  are  similarly 
defined  is  the  centre  of  gravity  of  unequal  particles. 


§  XVI.]  CENTRE  OF  GRAVITY  OF  A   CONTINUOUS  BODY.  235 

The  second  member  of  equation,  (i),  that  is,  the  aggregate 
of  the  distances  multiplied  by  the  masses,  is  called  in  Mechanics 
the  Statical  Moment  of  the  total  mass  2m  with  respect  to  the 
plane  of  yz. 


The  Centre  of  Gravity  of  a  Continuous  Body. 

205.  The   property  of  the   centre  of  gravity  given  in  the 
preceding   articles     obviously   extends    to   continuous   bodies. 
That  is  to  say,  the  position  of  this  point,  which  is  sometimes 
called  the  Centroid,  is  determined  by  the  mean  distances  of  the 
particles  from  three  given  planes;  and,  when  given,  it  determines 
the  mean  distance  from  any  plane.      If  the  body  is  homogeneous  t 
that  is  to  say,  if  the  same  quantity  of  matter  is  contained  in 
equal  elements  of  volume,  the  mean  distances  are  the  ordinary 
arithmetical  means.      On  the  other  hand,  if  the  body.  is   not 
homogeneous,  its  density  at  a  given  point  constitutes  the  weight 
to  be  attributed  to  the  element  at  that  point,  supposing,  as  in 
Art.   135,  the   geometric    elements   to   be   all   equal.      In  any 
case,  the  variable  density  is  simply  used  as  a  factor  of  the  ele- 
ment both  in  finding  the  total  mass  'as  in  Art.  135)  and  in 
finding  the  statical  moment. 

206.  The  centre  of  gravity  of  a  homogeneous  solid  is  also 
called  the  centre  of  gravity  of  the  volume,  and  the  integral  of 
xdV  (the  factor  of  density  being  omitted)  is  called  the  statical 
moment  of  the  volume.      So  also  the  centre  of  gravity  of  a  thin 
homogeneous  plate  of  uniform  thickness  is  called  the  centre  of 
gravity  of  the  area,  and  the  integral  of  xdA  is  called  the  stati- 
cal moment  of  the  area.      Thus,  for  the  circle  x*  -j-  y*  =  a2,  the 
statical  moment  with  respect  to  the  axis  of  y  of  the  semicircle 
on  its  right  is 


[xdA=2  r*yd 


—  2 


236  MEAN    VALUES  AND    PROBABILITIES.     [_Art- 


Dividing  by  the  area  %7raz,  we  have,  for  the   abscissa  of  the 
centre  of  gravity  of  the  semicircle, 


x  = 


207.  One  or  more  of  the  coordinates  of  the  centroid  may 
be  obvious  from  considerations  of  symmetry.      For  example, 
the  centroid  of  a  plane  area  or  of  a  plane  curve  is  in  the  plane  ; 
that  of  a  straight  line  is  at  its  middle  point  ;  that  of  a  sphere  or 
of  an  ellipsoid  is  at  its  centre  ;  that  of  a  hemisphere  is  on  its 
central  radius  ;  that  of  a  right  cone  is  on  its  axis. 

Again,  separation  of  the  body  into  elements  whose  centroids 
are  known  may  lead  to  similar  results.  For  example,  an 
oblique  cone  can  be  separated  into  circular  elements  of  uniform 
thickness  whose  centroids  are  therefore  at  their  centres. 
These  points  are  all  situated  on  the  geometrical  axis,  therefore 
the  centroid  of  the  cone  is  also  on  that  axis.  In  like  manner, 
the  centre  of  gravity  of  any  triangle  is  upon  a  medial  line.  It 
follows  that  it  is  at  the  intersection  of  the  three  medial  lines. 
It  is  obvious  also  that,  if  three  equal  particles  be  situated  at 
the  vertices  of  any  triangle,  their  centre  of  gravity  will  be  at 
the  same  point.  Hence  the  distance  of  the  centre  of  gravity 
of  a  triangle  from  any  straight  line  in  the  plane  is  the  arith- 
metical mean  of  the  distances  of  the  vertices. 

208.  When  integration  is  required,  the  element  of  moment 
employed  may  be  the  moment  of  any  convenient  element  of 
the  mass  or  volume.      For  example,  in  the  case  of  a  hemi- 
sphere of  radius  a,  it  is  necessary  to  use  integration  in  finding 
the  distance  from  the  base.      We  may  take  for  element  of  vol- 
ume the  hemispherical  shell  of  radius  r  and  thickness  dr,  of 
which  we  have  already  found  in  Art.   198  the  mean  distance 
from  the  base  to  be  one-half  of  the  radius.      Therefore,  since 


§  XVI.]       SQUARED    DISTANCES  FROM  A    PLANE.  237 

the   volume  of  the   shell  is   mr*  dr,    its    moment   is 
Integrating,  we  have  for  the  hemisphere 


moment 


ra  Tta* 

=  n     rs  dr  =  —  . 
Jo  4 


Dividing  by  the  volume,  which  is  -f  TTtf3,  we  find  the  height  of 
the  centre  of  gravity  above  the  base  to  be  f«.  This  is  there- 
fore the  average  distance  from  the  base  of  all  the  points  within 
the  hemisphere. 

The  moment  might  be  found  as  the  result  of  a  simple  in- 
tegration, in  this  case,  also  by  using  the  circular  element  of 
volume  parallel  to  the  base.  The  method  employed  above  is 
particularly  adapted  to  a  sphere  of  varying  density  if  the 
density  is  a  function  of  the  distance  from  the  centre  only. 


Average  Squared  Distance  of  Points  from  a  Plane. 

209.  The  mean  square  of  the  distances  of  points  in  a  plane 
area  from  a  straight  line  in  the  area,  and  that  of  the  distances 
of  points  in  a  volume  from  a  fixed  plane,  or  from  a  fixed 
straight  line,  have  important  application  in  Mechanics. 

In  the  case  of  the  area  A,  if  we  denote  the  mean  squared 
distance  from  the  axis  of  y  by  .r2,  it  is  found  from  the  formula 


=    {(x*dydx, 


where  the  integration  extends  over  the  area  A .  The  second 
member  of  this  equation  is  known  as  the  moment  of  inertia  of 
the  area  with  respect  to  the  axis  of  y,  and  is  usually  denoted 
by  /.  TKe  distance  k,  such  that  k2  —  x~,  is  called  the  radius 
of  gyration  of  the  area  with  respect  to  the  same  axis ;  thus  the 
equation  becomes  Jf-A  =  I. 

210.   As  in  finding  the  statical   moment,  so  in  finding  the 
moment  of  inertia,  it  may  be  convenient  to  use  some   other 


238  MEAN    VALUES  AND   PROBABILITIES.     [Art.  2IO. 

element  of  area  in  place  of  dy  dx.  For  example,  in  finding 
the  moment  of  inertia,  with  respect  to  the  axis  of  y,  of  the  loop 
of  the  lemniscate  r2  =  <£  cos  2&,  of  which  the  area  is  found  in 
Art.  109,  we  use  the  polar  element  of  area  rdr  dQ.  The  dis- 
tance of  the  element  from  the  axis  is  x.=  r  cos  6.  Hence 


/  =  2  [4  [V3  cos2  BdrdB  =    -  [4cos2  20  cos2  0  </0. 

J  o  J  o  .      2  J  o 

Putting  0  —  2(9  (see  Art.  96),  this  becomes 


and  since,  as  found  in  Art.  109,  A  =  £#2,  &A  =  /gives 


48 

211.  In  the  case  of  a  solid  the  "moment  of  inertia  "  and 
'  '  radius  of  gyration  '  '  depend  upon  the  distances  of  the  par- 
ticles, not  from  a  plane,  but  from  a  given  straight  line  in 
space.  Thus,  referring  the  body  to  rectangular  coordinate 
planes,  if  r  denotes  the  perpendicular  upon  the  axis  of  z  from 
the  point  P  at  which  the  particle  m  is  situated,  the  moment  of 
inertia  for  the  axis  of  z  is 

/,  =  2m**  =  Mk\  ......     (i) 

Now  since  r*  =  X*  -j-  yz, 

-j-  2  my*  .....      (2) 


§  XVI.]    SQUARED  DISTANCES  FROM  A  STRAIGHT  LINE.   239 

Thus  the  moment  of  inertia  is  the  sum  of  two  such  integrals  as 
are  defined  in  the  preceding  article.  Dividing  equation  (2)  by 
M,  the  total  mass,  it  becomes 

%  =  **+? (3) 

212.  The  value  of  /can  often  be  found  directly  from  its 
definition  2mr*  instead  of  using  the  second  member  of  equation 
(2).  For  example,  to  find  the  moment  of  inertia  of  a  homo- 
geneous cylinder  of  radius  a  and  height  h  with  reference  to 
its  geometrical  axis.  The  integral  expression  for  the  entire 
moment  of  inertia  of  a  volume  is 


/ 


=  \9* 


where  dV  is  an  element  such  that  r  is  the  distance  of  every 
particle  in  it  from  the  line  or  axis  of  reference.  In  other 
words,  dV  is  the  cylindrical  element  or  differential  of  volume, 
which  in  Art.  120  was  seen  to  be  particularly  adapted  to  find- 
ing the  volume  of  solids  of  revolution.  In  the  present  case, 
the  height  of  the  cylindrical  surface  is  h  and  its  circumference 
is  2nr.  Thus  dV '=  2nhr  dr,  and  dl =  2nhrzdr;  whence 


f* 

/=  2nk\  r*dr  = 

Jo 


and,  since  V  —  Tf/ia2,  the  corresponding  value  of  $  is  ^a2. 

213.  As  we  should  expect,  this  value  of  &  is  independent 
of  h\  it  therefore  applies  to  a  circular  lamina;  in  other  words, 
it  is  the  value  of  &  for  a  circle  with  reference  to  its  axis,  that 
is  to  say,  a  line  passing  through  its  centre  and  perpendicular 
to  its  plane. 

The  symmetry  of  the  cylinder  shows  that,  referring  its  base 
to  rectangular  diameters  as  axes  of  x  and y,  2mxz  and  2  my* 


240  MEAN    VALUES  AND   PROBABILITIES.     [Art.  213. 

must  be  equal.  Therefore,  x~  =  y*\  hence  from  equation  (3), 
Art.  211,  X*  is  one-half  the  value  of  &  found  above,  that 
is  to  say  \a2. 

This  is  the  value  of  kl  for  a  circular  area  with  reference  to 
a  diameter;  but  it  must  be  noticed  that  it  is  not  a  value  of 
k1  for  the  cylinder.  For  that  figure,  the  value  of  ky  must  be 
determined  by 

Mk*  =  Iy  =  2mx*  -f  2mz*, 

in  which  z  is  a  coordinate  in  the  direction  of  the  geometrical 
axis.  See  Ex.  23  below. 

More  extended  discussions  of  these  integrals  and  the  rela- 
tion between  them  for  different  axes  will  be  found  in  Treatises 
on  Mechanics. 

Examples  XVI. 

1.  Find  the  mean  value  of  the  tangent  of  an  arc  between  o  and 

45°-  2l°g*  2  _ 

~^~     ~  -441' 

2.  Find  the  mean  value  of  tan""1  x  when  o<x<  i. 

n        i 

-  --log  2  =  .439. 

3.  A  number  n  is  divided  at  random  into  two  parts.     Find  the 
mean  value  of  the  product.  £«J. 

4.  Find  the  mean  value  of  the  square  of  one  of  two  parts  of  the 
line  a  taken  at  random.  l^ 

5.  Find   the  mean   value    of   the  square  root  of  one   of   two 
random  parts  of  a.  f  \/a. 

6.  Find  the  mean  value  of  the  chord  drawn  from  a  point  on  the 
circumference  of  the  circle  whose  radius  is  a,  all  directions  of  the 
chord  being  equally  probable.  40 

7t' 

7.  What  is  the  average  distance  of  points  within  a  sphere,  radius 
a,  from  the  centre  ?  fa. 


§  XVI  ]  EXAMPLES.  241 

8.  The  half  of  a  right   cone  of  height  h  and   radius   of   base  b 
stands  on  its  triangular  surface  as  base  ;  what  is  the  mean  ordinate 
of  the  curved  surface  ?  Tib 

~6' 

9.  What  is  the  mean  height  above  the  base  of  a  point  in  the  sur- 
face given  in  Ex.  8  ?  46 

3* 

jo.  Find  the  mean  value  of  the  chord  AB  of  a  sphere,  radius  a, 

drawn  from  a  fixed  point  A  on  the  surface,  (»)  when  all  positions 
of  B  on  the  sphere  are  equally  probable  ;  (ft)  when  all  directions 
(see  Art.  199)  of  AB  are  equally  probable.  (a)  \a  ;  (fi)  a. 

1 1.  Show  that,  when  a  space  is  described  by  a  point  with  variable 
velocity,  the  mean  velocity  at  a  random  instant  is    the    ordinary 
"  average  velocity,"  or  whole  space  divided   by  the  whole    time. 
Show  also  that  for  a  body  falling  from  rest  (where  v  —  g t,  v*  —  2gs) 
this  mean  velocity  is  \  of  the  final  velocity,  while  the  mean  velocity 
of  passing  a  random  point  is  f  of  the  final  velocity. 

12.  Regarding  the  earth  as  a  sphere,  what  is  the  average  latitude 
of  a  point  in  the  northern  hemisphere  ?  32°  41'  15". 

13.  Given  that  the  horizontal  range  of  a  projectile  fired  with  the 

V 
velocity    V  and  the  angle  of  elevation  0  is  /t1  =  —  sin  2$,  show 

o 

that  the  mean  range  of  a  projectile  fired  in  a  random  direction  is 
two-thirds  of  the  maximum  range. 

14.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
parabola  y*  =  4mx  and  the  double  ordinate  corresponding  to  the 

abscissa  a.  „ 

*  =  £*. 

15  Find  the  centre  of  gravity  of  the  area  between  the  semi- 
cubical  parabola  ay*  =  x*  and  the  double  ordinate  which  corre- 
sponds to  the  abscissa  a.  —  _  K 

x  —  7  ft- 

1 6.  Find  the  coordinates  of  the  centre  of  gravity  of  the  area 
between  the  axes  and  the  parabola 


(-} 

W 


y= 


242  MEAN    VALUES  AND   PROBABILITIES.   [Ex.  XVI. 

17.  Find  the  centre  of  gravity  of  the  area  between  the  cissoid 
yt(a  —  x)  =  x*  and  its  asymptote.  ~x  =  fa. 

1  8.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
positive  directions  of  the  coordinate  axes  and  the  four-cusped  hypo- 
cycloid 

292  - 

apf  -f-  j*  =  rff.  _     _      2560 

x=  y=  —  *  — 


19.   Given  the  cycloid, 

y  =  a(i  —  cos  ^>),  x  =  a($>  —  sin  ip), 

find  the  distance  of  the  centre  of  gravity  of  the  area  from  the  base. 


20.  Determine  the  distance  from  the  base  of  the  centre  of  grav- 
ity of  a  hemisphere  when  the  density  varies  inversely  as  the  distance 
from  the  centre.  %a. 

21.  Find  the  mean  squared  distance  from  the  base  for  the  par- 
ticles of  a  homogeneous  hemisphere.  oP  =  ^a2. 

22.  Determine  the  radius  of  gyration  of  a  sphere  for  a  diameter 
by  means  of  the  result  of  Ex.  21,  and  also  directly  by  the  method 
of  Art.  212.  k*  =  \a\ 

23.  Determine  the  radius  of  gyration  of  a  cylinder  with  respect 
to  a  diameter  of  the  base.  J?  =  \a^  +  i//2- 

24.  Find  the  radius  of  gyration  of  the  area  between  a  parabola 
and  a  double  ordinate  2y  perpendicular  to  its  axis,  with  respect  to  a 
perpendicular  to  its  plane  passing  through  its  vertex. 

tf  =   JX*  +  ^y\ 

25.  Determine  the  radius  of  gyration  of  a  paraboloid  about  its 
axis,  b  denoting  the  radius  of  the  base.  &  =  $IP. 

26.  The  cardioid 

r  =  a  (i  —  cos  ff) 

revolves  about  the  initial  line;  determine  the  radius  of  gyration  of 
the  solid  formed  with  respect  to  this  line.  tf  =  |f  a2. 


§  XVII.]    MEAN  DISTANCES  FROM  A    FIXED    POINT.         243 

XVII. 

Mean  Distances  from  a  Fixed  Point. 

2I4-.  We  have  seen  that  the  mean  value  of  a  variable  de- 
pends not  only  upon  the  restrictions  imposed  in  the  question 
upon  the  values  of  the  variable,  but  upon  the  distribution  of 
the  admissible  values.  This  distribution  is  analytically  ex- 
pressed in  the  selection  of  the  element  employed,  and  the 
restrictions  are  expressed  by  the  limits  of  integration.  For 
example,  in  finding  the  mean  distance, 
from  a  fixed  point  0,  of  the  points  on  a 
straight  line  AB  of  limited  length,  the 
variable  is  OP,  Fig.  38,  and  the  distribu- 
tion of  the  values  (expressed  in  words  by 
the  statement  that  P  falls  at  random  be- 
tween the  limits  assigned),  is  provided  for 
by  selecting  the  element  dy,  where  y  is  the 
distance  of  P  from  a  fixed  point  of  the  line.  Taking  for  this 
point  A  the  foot  of  the  perpendicular  OA  =  a  from  the  point  O, 
the  expression  for  the  variable  in  question  is  OP  =  y(d*  -f-  jj/2). 
Thus,  if  A  and  B  are  given  as  the  limiting  positions  of  P,  the 
"whole  number  of  cases  "  is  represented  by  AB  =  b,  which 
is  the  integral  of  the  element  dy  between  the  limits  o  and  b. 
The  mean  distance  is  then  given  by  the  equation 

bM 


=       t/0a+/)  dy  =  -\b  ^(&+P}+a*  log 
Jo  ^  L_ 


see  formula  (Z),  p.   125. 

215.  As  a  particular  case  of  this  result,  putting  b  =  a,  we 
have  for  the  mean  distance  from  one  corner  of  a  square  of  a 
point  taken  at  random  upon  either  of  the  opposite  sides 


244  MEAN    VALUES  AND    PROBABILITIES.     [Art.  21$. 

M  —  $a[  |/2   -f-  log  (I   -f-  |/2)]  =    I  .  1480, 

where  a  is  a  side  of  the  square.  It  readily  follows  that  this  is 
the  mean  distance  from  the  centre  of  a  point  on  the  perimeter 
of  a  square  of  side  2«,  and  therefore  0.574^  is  the  same  mean 
when  the  side  is  a. 

Again,  putting  b  =  a  1/3,  so  that  the  angle  AOB  is  60°,  we 
find,  for  the  mean  distance  from  the  centre  of  a  point  on  the 
perimeter  of  an  equilateral  triangle  circumscribing  the  circle 
whose  radius  is  a, 

Jf =.«[i -fit's  toff  (V3+*)]i 

or,  if  s  is  a  side  of  the  triangle, 

M=  Ty[2/3  +  log  ( 4/3  +  2)]  =  0.398*. 

216.  If/'  in  Fig.  38  is  taken  at  random  within  the  rect- 
angle with  vertex  at  O  and  sides  a  and  b,  the  mean  distance 
is  the  value  of  M  in  the  equation 


abM '  •=.  d(x*  ~\~  y*}dy  dx,    .      .     .      .      (i) 

J  o  J  o 

in  which  the  "  number  of  cases  "  is  represented  by  the  area  of 
integration  ab.     Here,  the  result  of  the  first  integration  would  be 


which  gives  a  troublesome  form  to  the  second  integration. 

But,  if  we  replace  the  upper  limit  b  by  mx,  where  m  is  con- 
stant, this  becomes 


-f  w>)-{-log[>  +|/(i+  m*}-]\**  =  Kx\     .      (2) 


§  XVII.]    MEAN  DISTANCES  FROM  A    FIXED    POINT.         245 


and  the  second  integration  becomes  very  simple.  In  doing  this, 
we  integrate  over  a  triangle  instead  of  a  rectangle.  Thus  if, 
in  Fig.  38,  b  =  ma,  the  mean  distance  from  0  of  a  point  within 
the  triangle  OAB  is  given  by 


-mtPM. 

2 


ftt  r 
.  = 

J  o  J  o 


f 

= 

J 


Kx*dx  = 


Kaz 


(3) 


2Ka 


whence  M.  =  -  — ,  where  K  is  the  constant  defined  in  equa- 
Zm 

tion  (2). 

It  is  obvious  that  the  value  of  the  integral  in  equation  (i) 
can  thus  be  obtained  by  dividing  the  field  of  integration  into 
two  parts.  Again,  because  the  two  parts  of  the  rectangle  ab 
are  equal,  if  M2  is  the  value  of  the  mean  in  the  second  part,  we 
shall  have  M=  \(M^  -f  J/2). 

217.  The  element  in  the  double  integral  of  the  area,  which 
here  represents  the  number  of  cases,  is 
dy  dx,  which  we  may  denote  by  d~*A.  In 
the  single  integral,  the  element  is  the  result 
of  integrating  this  for  y\  that  is  dA  =  y  dx 
taken  between  certain  limits  for  y.  Thus, 
in  integrating  over  the  half  square  OA  C  in 
Fig.  39,  it  is  dA  =  x  dx.  Now,  if  M0  de- 
notes the  mean  value  of  the  variable  in 
question  for  this  element,  the  result  of  the  first  integration  of 
the  expression  for  the  aggregate  must  equal  M0dA ;  hence, 
M  being  the  final  mean,  we  have 


39- 


M-A  — 


(i) 


M0  is  here  a  function  of  the  independent  variable  x,  and  M0dA 
may  be  called  the  aggregate  of  the  new  cases  introduced  when 
x  is  changed  to  x  -j-  dx. 


246  MEAN    VALUES  AND    PROBABILITIES.     [Art.  2 1 7. 

This    equation    is    frequently    useful    in    extending  results 
already  found.      Thus,  in  Fig.  39,  M0  is  the  mean  value  of  OP 
for  a  side  opposite  to  O  of  the  square  whose  side  is  x;  hence 
by  Art.  215,  M0  =  1.148*.      Therefore,  by  equation  (i), 


whence 

M—  f  X  1.148*=  .765*. 

Mean  Distances  between    Two   Variable  Points. 

218.  When  each  of  two  points  is  taken  at  random  upon 
a  fixed  line,  the  distance  between  them  is  a  function  of  two 
variables,  since  the  position  of  each  point  is,  in  that  case,  de- 
termined by  a  single  variable.  For  example,  if  each  of  the 
points  lies  upon  the  circumference  of  the  same  circle  of  radius 
a,  it  is  determined  by  an  angle  at  the  centre  measured  from 
some  fixed  radius  OA.  The  problem  is,  in  this  case,  to  find  the 
mean  value  of  a  chord  under  certain  re- 
strictions. If  there  are  no  further  restric- 
tions, so  that  each  end  of  the  chord  is 
\A  equally  likely  to  fall  at  any  point  of  the 
circumference,  it  is  obvious  by  symmetry 
that  we  may  assume  one  end  to  be  at  the 
fixed  point  A,  and  the  other,  B,  to  fall  at 
FIG.  40.  random  upon  the  semicircumference  ABD 

in  Fig.  40.     This  restricts  the  number  of  cases  to  n,  and  we 
have  A  B  =  2a  sin  i#.     Therefore 


,_ 
M= 


2a  t*  . 
=—\   s 
7t  J0 


is  the  mean  value  of  a  chord  when  all  positions  of  the  extremi- 
ties are  equally  probable. 


§  XVII.]        MEAN  DISTANCES  BETWEEN  POINTS.  247 

219.  Now  suppose  that  we  require  the  mean  value  Mv  of 
the  chord  BC  which  cuts  a  fixed  diameter  AD.  Then  the 
two  variables  0  =  A  OB  and  0  =  AOC'in  Fig.  40  determine 
the  positions  of  the  extremities,  and  each  is  restricted  to  values 
between  o  and  TT.  The  number  of  cases  is  now  ?r2,  and  the 
value  of  BC  is  2a  sin  £(0  +  0)-  Therefore 


n»  f  -ie=r 

sin  $(0  +  <p}dOd(f>  =  —  4^     cos  $(0+0)         d<j> 
-      O  Jo  —  '0  =  0 

r  r  n* 

=  40    (cos$0+sin$0)*/0=8tf   sin$0—  cos£0     =160; 


whence  M,  = 


n* 


220.  When  the  lines  on  which  the  random  points  fall  have 
a  common  part  (in  which  case  zero  will  be  a  possible  value  of 
the  distance),  care  must  be  taken  that  the  expression  for  the 
distance  does  not  become  negative.  For  example,  if  both  ex- 
tremities of  the  chord  B  and  C'  fall  at  random  in  the  upper 
semicircumference  in  Fig.  40,  the  expression  for  the  chord  is 
BC'  =  2a  sin  £(0  —  0),  which  has  a  negative  value  when 
0  <  0.  The  aggregate  formed  by  integrating  over  the  range 
o  to  TT  for  both  variables  would  obviously  be  zero  in  this  case. 
What  we  require  is  the  mean  of  the  positive  value  of  this  ex- 
pression. This  limits  the  number  of  cases  to  i^2,  namely 
those  in  which 

TT>  0  >  0  >  o; 

thus,  if  we  integrate  first  for  0,  its  limits  are  o  and  0.  Denoting 
the  mean  in  this  case  by  Mz ,  we  have  for  the  mean  value  of 
the  chord  which  does  not  cut  a  given  diameter 


248  MEAN    VALUES  AND    PROBABILITIES.    [Art.  22O. 


pr  ,+ 

=  2a\       si 

J  o  J  o 

=  4a\  (i  — 

J  o 


cos 


Therefore        M2  =          ~       . 

TT 

The  whole  number  of  chords  which  do  not  cut  the  diameter 
through  A,  inclusive  of  those  which  lie  below  the  diameter,  is 
7t2,  the  same  as  that  of  the  chords  which  do  cut  the  diameter. 
Therefore  the  mean  value,  when  all  restrictions  are  removed, 
is  M  =  \(M^  -\-  M^.  Accordingly,  the  values  found  in  this  and 
the  preceding  article  are  verified  by  the  value  of  M  found  in 
Art.  218. 


Mean  Distances  Connected  with  a  Sphere. 

221.  In  finding  the  mean  distance  of  the  point  B  taken  at 
random  within  the  volume  of  the  sphere 
whose  radius  is  a  from  the  point  A  taken 
at  random  upon  the  surface,  we  may 
obviously  take  A  as  fixed.  Taking  this 
point  as  the  pole  in  Fig.  41  (which  rep- 
resents a  section  through  A ,  B  and  the 
FIG.  41.  centre),  the  distance  is  the  radius  vector 

of  the  point  By  and  the  number  of  cases  is  the  volume,  |7r^3, 
of  the  sphere.  In  finding  the  aggregate  of  the  r's,  we  may 
avoid  triple  integration  by  taking  for  dV,  as  in  Art.  146,  the 
volume  generated  by  the  element  of  area  rdrdQ  when  the 
figure  revolves  about  the  initial  line.  Thus 

dV  =  27rrjsin  QdrdS\ 


§  XVIL]    DISTANCES   CONNECTED     WITH  A    SPHERE.        249 

then  the  mean  value  of  r  is  determined  by 


Oza  cos  6 
r3  dr  sin  0  dd 
o 


•  TC  fa 

.    =  —     ( 

2  Jo 


2a  cos  &y  sin  8  dd  =  f  / 

**  J  o 

Therefore  J/=  £#. 

222.  This  result  may  be  extended  to  the  case  of  two 
points  both  taken  at  random  within  the  sphere  by  the  method 
of  Art.  217,  which  applies  equally  well  when  the  total  number 
of  cases  is  not  represented  by  an  area.  For,  suppose  N,  the 
total  number  of  cases  to  be  expressed  in  terms  of  a  single  in- 
dependent variable,  say  r,  so  that  dN  is  the  number  of  new 
cases  introduced  when  r  is  changed  to  r  -(-  dr.  Then,  if  M9  is 
the  mean  value  for  these  new  cases,  the  final  mean  M  will  be 
determined  by 

-Jo-- 

In  the  present  problem,  when  both  points  are  taken  at  ran- 
dom within  the  sphere  whose  radius  is  r,  the  whole  number  of 
cases  is  N=  (^Trr5)2  =  V-wV,  whence  dN  =  J/?r2.  6r*dr.  The 
"  new  cases,"  of  which  this  is  the  number,  are  plainly  those 
in  which  one  of  the  points  is  on  the  surface  of  the  sphere  and 
the  other  anywhere  within;  hence,  by  the  preceding  article, 
Mn  =  \r.  Therefore 


:  l/7r2.6.f    r*dr\ 

J  o 

whence  M  =  ||#.* 

*It  may  be  noticed  that  in  this  process  if  N when  expressed  in  terms  of  the 
single  variable  is  of  the  form  kz"  and  Mo  is  of  the  form  czm ,  the  differentiation  of 


250  MEAN    VALUES  AND    PROBABILITIES.    [Art.  223. 


Random  Parts  of  a  Line  or  Number. 

223.  The  division  of  a  line  (which  may  be  taken  to  repre- 
sent a  number  regarded  as  a  continuous  quantity  capable  of 
indefinite  subdivision)  into  three  parts  at  random  involves  the 
random  fall  of  two  points  of  division.     Each*  point  of  division  is 
equally  likely  to  fall  at  any  position  on  the  line.     Let  AB  be 
the  line,  and  R  and   5  the  points   of  division;  then,  putting 
AR  =  x  and  AS  =  y,  it  is  to  be  noticed  that  x  is  not  one  of 
the  three  parts  unless  y  >  x.     If  this  restriction  is  made,  the 
parts  are 

x,       y  —  x,       a  —  y. 

Without  the  restrictions,  the  mean  value  of  x  would  of  course 
be  %a,  and  the  mean  value  of  y  —  x  would  be  zero,  since  for 
every  case  in  which  y  —  x  has  a  positive  value  there  is  a  case 
in  which  it  has  an  equal  negative  value.  But,  when  the  re- 
striction is  made,  the  mean  value  of  y  —  x  is  the  same  as  the 
mean  distance  of  two  points  taken  at  random  on  a  straight 
line,  in  finding  which  it  is  necessary  to  exclude  negative 
values,  as  in  Art.  220. 

224.  That  there  is  no  distinction  in  kind  between  the 
three  parts  thus  symbolized  is  clearly  seen  if  we  imagine  the 
line  bent  into  the  circumference  of  a  circle  upon  which  three 
points  fall  at  random.     No  change  is  made  by  taking  one  of 
the  points  as  fixed. 

It  follows  that  the  mean  values  of  the  three  parts  are  equal. 
Again,  since  the  sum  of  the  corresponding  terms  which  enter 

N  introduces  the  factor  «,  and  the  integration  introduces  the  divisor  m-\-n;  so 
that  the  relation  between  M  and  M0  for  the  same  final  value  of  z  is 


Mo. 


Thus,  in  Art.  217,  we  ha_  M—  \MQ;  in  the  present  case,  we  have  M  =  §M\ 


§  XVII.]    RANDOM  PARTS   OF  A    LINE    OR   NUMBER.          25  I 


the  three  aggregates  in  each  "  case  "  (or  mode  of  division)  is 
a,  the  sum  of  the  aggregates  is  a  multiplied  by  the  whole  num- 
ber of  cases.  Therefore  the  sum  of  the  means  must  be  a,  and 
each  mean  is  \a. 

225.   It  is  sometimes  useful  in  problems  involving  two  in- 
dependent variables,  like  the  present,  to      _  >Q 
represent  them  by  the  coordinates  of  a 
point,  and  to  consider  the  area  of  integra- 
tion as  in  Art.  128.      Thus,  in  Fig.  42, 
let  the  x  of  Art.  223  be  the  abscissa,  OR, 
and  y  the  ordinate  of  a  point  P  referred 
to    the    rectangular  axes   OA  and   OB. 
Then  the  condition  y  >  x,  while  each  is 
positive   and  less  than  a,  restricts  P  to  FlG-  42. 

the  triangle  OBC.  It  is  now  obvious  that  the  mean  value  of 
x  is  the  same  as  ~x,  the  abscissa  of  the  centre  of  gravity  of  the 
triangle  OBC,  which  is  %a,  agreeing  with  the  conclusion  in 
the  preceding  article. 

Again,  the  mean  value  AT  of  x*  is  given  by 


= 

J 


whence  M  =  \cfi.  This  is  in  fact  the  square  of  the  radius  of 
gyration  (Art.  209)  of  the  area  of  integration  OBC  with  refer- 
ence to  the  axis  of  y. 

226.  If  we  make  a  real  distinction  between  the  parts,  for 
instance,  if  we  let  x  represent  the  smallest  of  the  three  parts, 
the  area  of  integration  is  still  further  restricted.  Thus,  if  the 
three  parts  in  Art.  223  are  in  order  of  magnitude,  a  —  y  being 
the  greatest,  the  conditions  are  completely  expressed  by 

o  <  .r  <  j  —  x  <  a  —  y. 
Here,  the  first  inequality  shows  that  P  is  on  the  right  of  the  line 


MEAN   VALUES  AND    PROBABILITIES.    [Art.  226. 


x  =  o,  in  Fig.  43,  or  OB.  The  second  shows  that  P  is  on  the 
left  of  the  line  x  •=.  y  —  x,  or  y  =  2x, 
which  is  the  line  OD,  joining  O  to  D 
the  middle  point  of  BC '.  The  third 
inequality  shows  that  P  is  below  the 
line  2y  =  a  -j-  x,  which  is  the  line  CE, 
where  E  is  the  middle  point  of  OB  in 
the  figure.  Thus  P  is  restricted  to  the 
triangle  OEF.  The  intersection  F  of 
the  two  lines  is  the  point  (l.a,  \a). 
The  mean  value  of  x  is  therefore  the 


FIG.  43. 


abscissa  of  the  centre  of  gravity  of  this  triangle,  namely  \a. 
The  mean  value  of  y  is  the  ordinate  of  the  same  point,  which 
by  Art.  207  is  the  mean  of  the  ordinates  of  the  vertices,  that 
is  \(^a  -f-  \a]  =  -^a.  Hence  we  have,  for  the  mean  values  of 
the  least,  middle  and  greatest  parts, 

la,        Atf       and       J4#. 


Random  Division  into  n  Parts. 

227.  If  a  number  a  is  divided  into  n  parts  at  random,  n  —  r 
of  these  parts  may  be  taken  as  independent  variables  x,  y,  z 
etc.,  subject  to  the  condition  that  each  shall  have  a  positive 
value  and  that  their  sum  shall  not  exceed  a ;  thus, 

x  -\-  y  -\-  z  -f-  .  .  .  <  a. 

The  number  of  cases  is  now  represented  by  the  definite  inte- 
gral 

~  i — x  fa—x—y 

.  .  .dzdy  dXy 


fa  fa — x  fa 
J o*-o        J o 

which  involves  n  —  I  integral  signs. 


§  XVII.]         RANDOM  DIVISION  INTO   U    PARTS. 

In  like  manner,  the  aggregate  of  the  wth  powers  of  a  part 
(or  numerator  of  the  fraction  representing  the  mean  value)  is 
the  result  of  integrating  the  product  of  the  element  in  this 
integral  by  the  mth  power  of  any  one  of  the  variables.  The 
integral  will  be  found  to  have  the  same  value,  whichever  of 
the  variables  in  the  above  order  of  integrations  we  employ; 
but  it  is  simplest  to  employ  the  variable  for  which  we  first 
integrate. 

228.  Integrals  of  this  form  are  readily  evaluated  by  the 
•  help  of  the  theorem  of  Art.  97.  Thus,  in  the  case  of  five  parts, 
we  have  to  evaluate  the  quadruple  integral 

na—xfa—x—y  fa — x—y—z 
wmdw dz dy dx .  .        .      (i) 
o        J  o  Jo 

The  value  of  the  first  or  inner  integral  is 

-±-(,-_  *_,_.,)-+.. 

Putting  ft  for  a  —  x  —  y,  the  upper  limit,  the  integral  next  to 
be  evaluated  is 


By  the  theorem  referred  to,  this  becomes 

i       r0  fim  +  2  (a  —  x  — 

„«£  -4-  T    _/-.  V 


m  -f  i  J0  (m  +  i\m  +  2)        (in  +  i)O  +  2)' 

In  like  manner,  putting  y  for  a  —  x,  the  result  of  the  next 


interation  is 


(m  +!)(**  +  2}]y  ~  (m  +  i)(m  +  2](m  -f  3)' 

and  finally 

~  (m  +  !)(*«  +  2)(m  -j-  3)(*»  -J-  4)' 


254  MEAN   VALUES  AND    PROBABILITIES.    [Art.  22G. 

229.  This  integral  is  the  aggregate  of  the  mth  powers  of 
a  part;  also,  putting  m  =  o,  it  gives  the  total  number  of  cases, 
namely  ^a*;  hence  we  have,  for  the  mean  value  of  the 
power  of  one  of  five  parts  selected  at  random, 


o+ 

This  gives  \a  for  the  mean  of  a  part  (agreeing  with  Art.  224), 
-j^a*  for  the  mean  square  of  a  part,  and  so  on. 

In  the  general  case  of  n  parts,  we  have  in  like  manner,  for 
the  whole  number  of  cases, 

N  = 


(n-  i)!' 
and,  for  the  mean  mth  power  of  a  part, 

(n  -  i}\ml 
M=7±       -*  -  ri*'*. 
(m  -f-  n  —  i  )  ! 

230.  The  mean  value  of  a  product  of  powers  of  two  or 
more  parts  can  be  readily  found  by  the  aid  of  the  equation 


r\ 

x\a-xjdx  =  -.  -  j— 


f 
\ 
J 

We  have  now  to  evaluate  the  definite  integral 

7  =  j  J  j  •  •  •  *'*l  '  •  •  <-'0  -  2x)'dxn_l  dxn  _2  .  .  .  dxlt     (2) 


in  which  the  n  —  i  variables  are  positive  quantities  subject  to 
the  condition  2x  <  a,  and  each  of  the  exponents  /,  q,  .  .  .  s 
is  zero  or  a  positive  integer.  Denoting  the  limit  in  the  first 
integration  by  a  (compare  Art.  228),  the  part  of  a  last  written 

*  See  EK.  VII,  16,  p.  128. 


§  XVII.]        RANDOM  DIVISION  INTO   H   PARTS.  25$ 

is  a  —  ~2x  =  a  —  ^«_,  ;  hence  the  first  integral  has  the  form 
given  in  equation  (i).  At  the  next  integration,  putting  ft  for  the 
upper  limit;  a  =  ft  —  xn~n  substituting,  the  integral  to  be 
evaluated  takes  the  same  form,  s  being  replaced  by  s  -f-  r  -f-  i, 
and  r  by  a.  new  exponent  ;  so  that  the  resulting  exponent  is 
of  the  form  s  -j-  r  -f-  t  -J-  2  .  So  also  at  each  step,  the  new 
value  of  s,  in  the  application  of  equation  (i),  is  the  sum  of  the 
exponents  used  and  the  number  of  preceding  integrations. 

Consider  now  the  numerical  factors  introduced  at  the  suc- 
cessive steps.  In  the  denominator,  the  new  factors  run  from 
s  -j-  i  to  the  new  value  of  s  ;  therefore  these  factors  constitute 
the  series  of  increasing  natural  numbers.  In  the  numerator, 
the  new  factor  introduced  is  the  factorial  of  the  new  exponent. 
Hence  we  have  finally,  for  the  value  of  /in  equation  (2), 

/_  -  p.  q.  .  .  .s.  -         «_l+/+^  +  ...+, 
(n-  i+p  +  q  +  ...+s)\ 

Dividing  by  the  value  of  TV,  Art.  229,  we  find  that,  when 
a  is  divided  at  random  into  n  parts,  the  mean  value  of  the 
product  of  the  /th  power  of  one  part  by  the  ^th  power  of 
another  and  so  on,  is 


231.  The  mean  value  of  the  least  part,  the  next  to  the  least 
part,  and  so  on  to  the  greatest  part,  can  be  found  without  in- 
tegration as  follows  :  *  Denote  the  least  part  by  a,  the  next 
by  a  -f-  /?,  the  next  by  a  -f-  ft  -f-  y,  and  so  on.  Then  a,  fi,  y, 
etc.,  are  all  positive,  and,  summing  the  parts,  we  have 

na-\-(n  —  \)ft  -f  (n  —  2]y  +  .  .  .  =  a. 

The  terms  in  the  first  member  may  now  be  regarded  as 
random  parts  into  which  a  is  divided.  Therefore,  the  mean 

*  See  Whitworth's  "Expectation  of  Parts"  in  "Choice  and  Chance  "  where 
a]  so  an  algebraic  proof  of  the  value  of  M  in  Art.  230  is  given. 


MEAN    VALUES  AND   PROBABILITIES    [Art.  231. 

a  a  a 

value  of  each   term  is  — ;  whence  —  ,    —.        — r  ,  etc.,  are  the 

n  n*     n(n  —  I ) 

mean  values  of  «,/?,...  The  mean  value  of  a  -f-  ft  is  obvi- 
ously the  sum  of  those  of  a  and  ft,  and  so  on.  Hence  we 
have,  for  the  several  mean  values, 

a    I 
n'n'' 
a  /i 
n  \n 
a  I  \ 


n\n    '   n  —  I 


These  results  will  be  found  to  agree  with  those  found  by  inte- 
gration, in  Art.  226,  for  the  case  n  =  3. 

Mean  Area  of  a  Triangle  with  Random  Vertices. 

232.  When  the  quantity  whose  mean  is  required  depends 
upon  the  position  of  two  variable  points,  the  element  or  dif- 
ferential of  the  "  number  of  cases  "  is  the  product  of  the  ele- 
ments upon  which  the  two  points  re- 
spectively fall.  For  example,  let  us 
find  the  mean  area  of  the  triangle  APQ, 
Fig.  44,  formed  by  joining  the  fixed 
point  A  on  the  circumference  and  two 
points  P  and  Q  taken  at  random  within 

^^ 

the  circle.      It  is,  for  this  purpose,  con- 
venient to  refer  the  circle  to  polar  co- 
ordinates, taking  A   for  the  pole,  and  the  tangent  to  the  circle 
for  the  initial  line.     The  equation  of  the  circle  is  then 

r  =  2a  sin  6. 
Denoting  by  r  and  0  the  polar  coordinates  of  P,  and  by  p  and 


§  XVIL]       TRIANGLES    WITH  RANDOM    VERTICES. 

<f)  those  of  Q,  the  quantity  whose  mean  is  required  is  the  area 
APQ  =  \rp  sin  (0  —  #). 

The  element  of  the  number  of  cases  is  now  the  product  of  the 
elements  of  area  upon  which  P  and  Q  fall,  that  is, 

d*N  =  rdrdepdpdfi. 

To  avoid  negative  values  of  the  expression  for  the  area,  we 
suppose  6  <  0,  as.  in  Art.  220,  which  limits  the  number  of 
cases  to  one  half  the  product  of  the  areas ;  thus 

m2«  sin  <t>  fia  sin  9 
rdrpdpdOdff)  =  \n*a*>. 
Q  JO 

Using  the  same  limits,  the  aggregate  of  the  areas  APQ  is 

sin  <6  f?a  sin  6 


Hence 


mza  sin  cp  Cm  sin  9 
r*  dr  p2  dp  sin  (0  -  ff)dd  d$. 
o  Jo 

(za\6  f*  f* 

=  ¥-.     I    sin3  0    sin3  0  sin  (0  —  B}ded(t>, 

I<5     Jo  Jo 

64.0*  fn  f  * 

M  —  -^j-J    sin3  0     (sin  0  sin8  ^  cos  6  —  cos  0  sin4  6)d6  d<!>. 


and 

9^  J0 


Since 


(sin  0  sin3  V  cos  0  —  cos  0  sin4 


=  J  sin5  0  —  cos  0(f0  —  f  sin  0  cos  0  —  1  sin3  0  cos  0), 
this  becomes 

J/=   — J  (2  sin8  0  —  30  sin3  0  cos  0 

-|-  3  sin4  0  cos2  0  -f-  2  sin8  0  cos2  0)^/0. 


258  MEAN    VALUES  AND    PROBABILITIES-    [Art.  232. 

* 

The  value  of  the  integral  in  this  expression  is 

7.5.3.10-          r-    sin4  0-1"       3  r" 

4sT^-22-l\  0^       +7      sin' 
5-O-4-2  2          L       4     Jo      4Jo 


6-4-2  2    '   ^8.6.4-22 

-  35^  ,    3^3j^  ^    ,   3^.    S^  _  20  +  9  +  6^  _  35^ 
64     '    24-22        16"^  64"  32  32' 

Therefore  J/=  — -9 =  —E—. 

97T2    32         36^- 

233.  From  this  result  we  can  readily  derive,  by  the  method 
of  Art.  222,  the  mean  area  of  the  triangle  when  all  three 
vertices  are  taken  at  random  within  the  circle.  For,  if  r  is  the 
distance  of  the  point  farthest  from  the  centre,  the  value  of  N 
in  terms  of  r  is  N  =  TrV6,  whence  dN  =6x?r5dr;  and  the  mean, 
MQ,  of  the  new  cases  (of  which  dN  is  the  number)  is,  by  the 


preceding  article,  ^—.       Hence,    if  M  denotes  the  required 
mean  when  the  radius  is  a, 

whence 

6     35 
~  8  '$67ta 

Mean  Areas  Found  by  the  Method  of  Centroids. 

234.  The  theorems  proved  below  are  often  useful  in  find- 
ing the  mean  areas  of  triangles  with  random  vertices.  If  two 
vertices,  A  and  B,  of  a  triangle  are  fixed  and  the  third,  C, 
is  taken  at  random  upon  a  given  line  or  area  (£"),  its  area  is 


§XVIL]  MEAN  AREAS  BY  THE  METHOD  OFCENTROIDS.  2$Q 

\AB-p,  where/  is  the  perpendicular  from  C  upon  AB.  Sup- 
posing that  the  line  or  area  (£T)  lies  in  a  plane  containing  A 
and  B,  and  furthermore  that  the  line  AB  or  AB  produced  does 
not  cut  (C},  the  /'s  will  all  have  the  same  direction  and  their 
mean  value  will  be  the  perpendicular  from  G,  the  centre  of 
gravity  of  (C}.  Hence,  with  this  proviso,  the  mean  area  of 
the  triangle  ABC  is  that  of  the  triangle  ABG. 

For  example,  if  A  and  B  are  the  extremities  of  a  diameter 
of  the  circle  whose  radius  is  a,  and  C  is  taken  at  random  in  one 

o 

of  the  semicircles,  the  mean  area  of  ABC  is  —  ,  see   Art.  206. 

\* 

If  C  is  taken  at  random  on  one  of  the  semicircumferences,  the 

2tf2 
mean  area  is   - — ,  see  Art.  193.     By  symmetry,  these  are  also 

the  values  when  C  is  taken  at  random  on  the  whole  circle  and 
on  the  whole  circumference  respectively. 

235.  Next  suppose  that  B  is  not  fixed,  but  taken  at  random 
on  a  line  or  area  (Z?),  lying  in  the  plane  which  contains  (C)  and 
the  point  A,  and  also  that  the  line  AB  can  in  no  position  of  B 
cut  the  area  (£7);  in  other  words,  that  no  straight  line  through 
A  can  cut  both  (B}  and  (C).  Then  the  aggregate  of  the- 
areas  in  all  the  cases  in  which  B  has  a  certain  fixed  position  is 
(C}-ABG,  and  the  aggregate  of  all  values  of  ABG  when  B 
is  not  fixed  is  (B}.AFG,  where/7  is  the  centre  of  gravity  of 
(B].  Therefore  the  aggregate  of  all  values  of  ABC  when 
neither  C  nor  B  is  fixed  is  (€)(]&)•  AFG,  and  since  the  total 
number  of  cases  is  (£")•(/?),  the  mean  value  is  AFG. 

For  example,  let  the  vertex  A  of  a.  triangle  be  joined  to 
any  point  D  of  the  base,  and  let  points  P  and  Q  be  taken  at 
random  respectively  in  the  two  parts  ABD  and  ADC.  Then, 
denoting  by  M2  the  mean  of  the  triangles  APQ,  we  have 
M2  =  AFG,  where  F  and  G  are  the  centres  of  gravity  of  the 
two  parts.  Now,  from  the  known  position  of  the  centre  of  grav- 
ity of  a  triangle,  the  base  FG  is  f  of  %BC,  and  the  altitude  is  f 


260  MEAN   VALUES  AND   PROBABILITIES.    [Art.  235. 

of  that  of  the  triangle  ABC.     Hence  the  mean  area  required  is 


where  A  is  the  area  of  the  triangle  ABC. 
236.  Certain  extensions  of  this 
result  may  now  be  made.  In  the  first 
place,  let  Mbe  the  mean  area  of  APQ 
when  P  and  Q  are  each  taken  at  random 

anywhere  in  ABC.      It  is  plain  that  for 
FIG.  45. 

different  triangles  Mis  proportional  to 

A.  Hence  putting  BD  —  x,  DC  =  y,  BC  =  a,  we  have, 
for  the  means  when  P  and  Q  both  fall  in  the  triangle  whose 
.base  is  x,  or  both  in  the  triangle  whose  base  is  y,  respectively 


Now  putting,  for  simplicity,  x  =  y,  we  have  for  the  mean 
when  P  and  Q  fall  on  the  same  side  of  AD,  M^  =  \M;  and  for 
the  mean  when  they  fall  on  opposite  sides,  as  found  above, 
M2  =  \A.  But  since  the  areas  of  the  parts  are  now  the  same, 
M  —  \(MV  -j-  M2} ;  whence  M=  \M^  therefore 

M  —  -£-A. 

237.  Next  let  the  triangle  have  a  fixed  vertex  on  a  side  of 
the  triangle  ABC,  say  at  D,  Fig.  45.  No  straight  line  through 
D  can  cut  both  of  the  areas  ADB  and  ADC,  therefore  the  con- 
•dition  given  in  Art.  235  is  fulfilled,  and  denoting  as  before  by 
jM2  the  mean  area  when  one  of  the  other  vertices  is  taken  at 
jrandom  in  each  of  the  triangles,  we  now  have  M2  =  DFG,  or 


Denoting  again  by  Mx  and  My  the  mean  areas  when  both 
of  the  other  vertices  are  taken  at  random  in  ABD  and  in  ADC 


§  XVII.]  MEAN  AREAS  BY  THE  METHOD  OF  CENTROIDS.  261 

respectively,  these  means  have  the  same  values  as  in  Art.  236, 
because  the  fixed  vertex  is  still  one  of  the  vertices  of  the  tri- 
angle in  which  the  other  two  fall  at  random.  Hence 


,        ,  . 

a  27  a  27 

Now  denoting  by  MD  the  mean  area  when  the  other  tw<r 
vertices  are  taken  at  random  in  ABC,  we  notice  that  the  whole 
number  of  cases  is  made  up  of  those  in  which  both  variable 
vertices  fall  in  one  or  other  of  the  two  triangles,  and  those  in 
which  one  falls  in  each.  Of  these  classes  of  cases  the  means 
are  Mx  ,  My  and  Mr  Since  the  number  of  cases  in  which  P 
falls  in  ABD  and  in  ADC  respectively  are  proportional  to  the 
areas,  they  are  as  x:y.  Hence  the  numbers  of  cases  men- 
tioned above  are  as  x'*,  yz  and  2xy;  the  whole  number  being 
thus  represented  by  (x  -f-  yf  or  a2. 

It  follows  that 


My  -f- 
hence,  substituting  the  values  found  above, 


a       27 


or,  since  y  =  a  —  x, 


27«2W 

Now,  to  find  the   mean  value  when  D  is  taken  at  random 
upon  the  side  DC,  we  have  only  to  integrate   this  result  be- 


262  MEAN    VALUES   AND   PROBABILITIES-     [Art.  237. 

tween  the  limits  o  and  a.      Hence,  denoting   this  new  mean 
by  M0,  we  have 


—  "\ax-\- 2 c^]dx\ 
° 


whence 


=  —  (i  -44-2)  =  -. 

27  ^  9 


This  is  obviously  also  the  mean  when  one  vertex  is  taken  at 
random  upon  the  perimeter  and  the  other  two  at  random  within 
the  triangle  A. 

238.  The  mean  J/  when  all  three  vertices  are  taken  at 
random  within  the  given  triangle  may  now  be  found  by  the 
method  of  Art.  222.  Let  /  be  the  altitude  of  the  given  tri- 
angle, and  z  the  variable  altitude  of  a  similar  triangle.  Then 
the  area  of  the  variable  triangle  is  kz*,  where  k  is  a  constant, 
and  A  =  kjP.  The  number  of  cases  when  each  of  the  three 
points  is  taken  at  random  within  the  variable  triangle  is 
N  '  •=.  £?z6.  Hence  the  number  of  new  cases  introduced  when 
z  is  changed  to  z  -j-  dz  is  dN  =  6kzz5dz.  The  mean  MQ  of 
these  new  cases  has  just  been  shown  to  be  M0  =  \kz*.  There-^ 

fore  the  equation  MN  =    M0dN  gives 
&?M  =  \&  \  z1  dz 

J  o 

Whence 


239.  If  the  three  vertices  A,  J3,  C  of  a  triangle  be  taken 
at  random  respectively  on  the  given  lines  or  areas  (A),  (Z?), 
(C}  in  the  same  plane,  but  such  that  no  straight  line  can  cut 
all  three.,  it  can  be  shown  as  in  Art.  235  that  the  mean  area 


§  XVI I.]  EXAMPLES.  263 

of  the  triangle  ABC  is  that  of  the  triangle  EFG  whose  ver- 
tices are  the  centroids  of  (A),  (B)  and  (Q.  For  example,  \ve 
can  thus  find  the  mean  area  of  a  triangle  whose  vertices  are 
taken  at  random  in  any  three  arcs  of  the  same  circumference 
which  do  not  overlap. 

Again,  in  like  manner  it  may  be  shown  that,  if  four  points 
be  taken  at  random  each  within  a  given  volume,  the  mean 
volume  of  the  tetrahedron  of  which  they  are  the  vertices  is  that 
of  the  tetrahedron  whose  vertices  are  the  centroids  of  the  given 
volumes,  provided  there  is  no  plane  which  can  cut  all  four  of 
the  given  volumes. 


Examples  XVII. 

1.  A  point  falls  at  random  upon  a  triangle;  show  that  the  mean 
value  of  its  shortest  distance  from  the  perimeter  is  one-third  of  the 
radius  of  the  inscribed  circle.    Show  also  that  the  same  thing  is  true 
of  any  polygon  circumscribed  about  a  circle  and  of  the  circle  itself. 

2.  Give  the  corresponding  theorem  with  respect  to  a  tetrahedron 
and  a  sphere. 

3.  Two  points  are  taken  at  random  on  the  surface  of  a  sphere; 
what  is  their  mean  distance  in  a  straight  line,  and  also  along  the 
surface  ?  |0  ;     \na. 

4.  Find  the  mean  distance  of  a  point  taken  at  random  on  the  sur- 
face of  a  sphere  of  radius  a  from  the  farthest  and  also  from  the 
nearest  of  the  extremities  of  a  fixed  diameter. 

|(4  -  |/2)«  ;     f  1/2  .  a. 

5.  Find  the  means  of  the  squares  of  the  distances  considered  in 
Ex.  4  ;  also  show,  &  priori,  that  their  arithmetical  mean  must  be 
2«a,  and  is  the  same  thing  as  the  mean  of  the  squared  distance  of 
two  random  points  on  the  surface.  30";    a*. 

6.  A  number  a  is  divided  at  random  into  three  parts;  find  the 
mean  value  of  their  product.  sW- 


264  MEAN   VALUES  AND  PROBABILITIES.  [Ex.  XVII. 

7.  Find  the  mean  value  of  the  product  of  two  parts  when  a  is 
divided  at  random  into  four  parts.  sV*2- 

8.  Find  the  mean  distance  of  all  points  within  a  circle  of  radius 
a  from  a  fixed  point  on  the  circumference.  323 

~W 

9.  Determine  the  average  distance  between  two  poirts  taken  at 
random  within  the  circle.     See  Art.  222.  128*2 

45^' 

10.  Find  the  mean  distance  of  a  point  within  a  cone  of  height  h 

and  semi-vertical  angle  a  from  the  vertex.  sec***—  i 

-  i  —  h. 
2  tan  or 

IT.  Find  the  mean  distance  of  a  point  taken  at  random  within 

a  hemisphere  from  the  opposite  pole.         a  ,  ,  . 

—  (53-16  fa)  =1.519* 

12.  Determine  the  mean  value  of  a  chord  which  cuts  a  pair  of 
perpendicular  diameters  of  the  circle  whose  radius  is  a. 

32(2  —  ^2) 
-  —5  --  a  =  1.900. 

13.  Find   the  mean  value   of  the  product  of  two  of  the  three 
random  parts  of  a,  and  the  mean  value  of  the  cube  of  this  product. 


12  '         560 

14.  A  number  a  is  divided  at  random  into  three  parts;  find  the 
mean  square  of  the  least  part,  of  the  middle  part  and  of  the  greatest 
part.  a"  ^      iga*         85  a1 

54'      216  '       216  * 

15.  Find  the  mean  square  of  the  least  of  n  random  parts  of  a; 
also  the  mean  /Hh  power.  20*  (n—\)\p\a? 


16.  Find  the  mean  value  of  the  square  of  the  next  to  the  least  of 
n  random  parts  of  a.  2(37*"  —  3«  -f-  i)      , 


§XVIL]  EXAMPLES.  265 

17.  Find  the  mean  area  of  the  right  triangle  whose  hypothenuse 
is  c,  all  values  of  either  acute  angle  being  equally  probable.       c 

27t' 

18.  Find  the  mean   area  of  the  triangle  formed  by  joining  the 
centre  and  two  random  points  within  the  circle  whose  radius  is  a. 

vL 

97f' 

19.  Two  radii  are  drawn  at  random  from  O,  and  P  and  Q  are 
taken  at  random  upon  the  radii;  find  the  mean  area  of  OPQ.      a* 

47T* 

20.  P  and  Q  are  taken  at  random  one  in  each  of  the  semicircles 
into  which  the  diameter  AB  —  20  separates  a  circle  ;  find  the  mean 
area  of  APQ.  4«2 

3*' 

21.  If  A  is  a  fixed  point  of  the  circumference,  and  P  and  Q  are 
taken  at  random  as  in  Art.  232,  show  that  in  £•  of  the  whole  number 
of  cases  the   centre   O  is  within  the  triangle  APQ.     Denoting  by 
J/0  the  mean  area  in  these  cases,  and  by  J/,  the  mean  in  the  remain- 
ing cases,  thence  derive  the  values  of  M0  and  J/,  by  means  of  the 
results  of  Art.  232  and  example  20  above. 


22.'  If  three  points  are  taken  at  random  in  a  circle,  find  the  mean 
area  of  the  triangle  of  which  they  are  the  vertices  when  the  centre 
is  within  the  triangle,  and  also  when  it  is  not. 

M  _37<     M        iia* 

jrj  o  —          —          J.tA      ' . 

247T  247T 

23.  The  circumference  of  a  circle  is  divided  into  the  three  parts 
2aac,  2a/3,  2ay,  and  a  point  is  taken  at  random  in  each  part  ;  show 
that  the  mean  area  of  the  triangle  of  which  they  are  the  vertices  is 

no1  sintf  sin/?  siny 
zafiy 


266  MEAN    VALUES  AND  PROBABILITIES.     [Art.  240. 

XVIII. 

The  Measure  of  Probability. 

240.  When  it  is  unknown  whether  an  event  has  or  has 
not  happened  (or  in  the  case  of  a  future  event,  whether  it  will 
or  will  not  happen),  the  probability  which  we  attribute   to   it 
depends  upon  the  amount  of  knowledge  we  have  with  respect 
to  its  causes.      In  order  to  give  a  numerical  value  to  the  prob- 
ability, there  must  be  a  certain   set   of  elementary  events  or 
cases    which,    with    our    present    knowledge,    we    regard    as 
equally   probable.       In  a  certain    number    of  these    possible 
cases,  the  event  in  question  happens;  these  cases  are  said  to 
be  favorable  to  the  event,  while  the   others  are   unfavorable. 
Then  the  ratio  of  the  number  of  favorable  cases  to  the  whole 
number  is    taken    as    the    measure  of   the  probability    of  the 
event. 

For  example,  in  a  question  of  throwing  dice,  the  cases  which 
(knowing  nothing  to  the  contrary)  we  assume  to  be  equally 
probable  are  the  turning  up  of  the  different  combinations  of 
faces  which  can  be  selected  from  the  several  dice.  With  a 
pair  of  dice,  these  are  36  in  number.  Suppose  the  event  in 
question  to  be  that  of  making  a  throw  whose  sum  is  8 ;  then 
by  actual  count  there  are  found  to  be  five  favorable  cases  ; 
hence  the  probability  is  -£-$.  Since  there  are  31  unfavorable 
cases,  this  is  sometimes  expressed  by  saying  that  the  odds  are 
31  to  5  against  throwing  8. 

241.  When  the  magnitudes  of  the  quantities  concerned  in 
a  question  of  probabilities  are   continuous,  that  is   capable  of 
indefinite  subdivision,  the  whole  number  of  cases  is  found  by 
evaluating  an  integral,  'exactly  as  in  the  treatment  of  mean 
values,  and  the  enumeration  of  the  favorable  cases  is  effected 
in  like  manner. 


§  XVIII.]         THE  MEASURE    OF  PROBABILITY.  267 

For  example,  suppose  that  two  events   are  known  to  have 
occurred  within  the  same  year,  and  noth- 

ing more  is  known;   required  the  prob-    ~  -  3L,  -  !L  -  -* 
ability  that  the  interval  of  time  between 
them  shall  exceed  a  certain  fraction  of 

a  year.  It  is  convenient  to  represent  the  year  by  the  straight 
line  OA,  Fig.  46;  then  the  instant  of  occurrence  of  each  event 
is  represented  by  a  point  taken  on  OA.  Denote  the  dis- 
tances from  the  beginning  of  the  year  by  x  and  y,  and  let 
x  =  (9Jf  be  the  greater.  If  only  one  point  were  concerned,  the 
equally  probable  cases  would  consist  of  the  falling  of  the  point 
X  upon  the  several  elements  dx  of  the  line  OA,  and  the  "  num- 
ber of  cases  '  '  would  be  represented  by  the  length  a  of  this  line. 
But  since  two  points  are  involved,  the  elementary  case  con- 
sist of  the  falling  of  X  and  Y  upon  a  particular  pair  of  elements 
dx  and  dy.  Since  we  have  assumed  y  <  x>  the  number  of  cases 
is  limited  to 


The  "event"  whose  probability  we  seek  is  that  the  distance 
YX  shall  exceed  a  given  quantity  c,  equal,  say,  to  OD  in  Fig. 
46.  There  are  no  favorable  cases  when  x  <^;  but,  for  each 
value  of  x  between  c  and  a,  favorable  cases  occur  whenever 
y  <  x  —  c.  Hence  the  number  of  favorable  cases  is 


ra  rx~c  ca 

dy  dx  =        (•*•""  c}dx  = 

J  c  J  o  J  c 


(a  - 


Dividing  by  the  whole  number,  we  have  for  the  probability  P 

p  =  (a~,c)*. 

It  may  be  noticed  that  it  is  more  convenient,  in  this  prob- 
lem, to  find  the  probability  of  the  distance  exceeding  c,  because 


268 


MEAN   VALUES  AND   PROBABILITIES.     [Art.  241. 


the  number  of  cases  in  which  it  is  less  than  c  can  not  be  ex- 
pressed by  a  single  definite  integral. 

Probabilities  Represented  by  A  reas. 

242.  When  two  variables  occur,  it  is  often  useful  to  rep- 
resent them  as  in  Art.  225  by  the  rectangular  coordinates  of  a 
point  which  thus  falls  at  random  within  an  area  which  repre- 
sents the  whole  number  of  cases.  The  number  of  favorable 
cases  will  then  be  represented  by  a  portion  of  this  area,  and 

may  often  be  found  without  integra- 
tion. Thus,  in  the  example  above,  the 
whole  number  of  cases  is  represented 
by  the  half  OA  C  of  the  square  in  Fig. 
47,  constructed  on  the  line  OA.  Tak- 
ing OD  =  c,  the  line  DE  parallel  to 
OC  is  the  locus  of  the  equation 
y  •=.  x  —  c.  Hence  the  favorable  area 
is  the  area  below  this  line,  that  is  the 
triangle  DAE. 

In  this  graphic  method  the  area  favorable  to  the  contrary 
event  is  exhibited  at  the  same  time ;  thus  ODEC  is  the  area 
favorable  to  a  distance  XY  less  than  c,  which,  as  remarked  in 
the  preceding  article,  is  not  expressible  by  a  single  integral. 

24-3.  The  method  is  particularly  useful  when  there  are  con- 
ditions which  still  further  restrict  the  favorable  area.  For 
example,  we  have  seen  in  Art.  223  that  the  points  X  and  Y 
in  Fig.  46  divide  the  line  a  into  three  parts  at  random.  Sup- 
pose now  that  we  require  the  probability  that  no  one  of  the  three 
parts  shall  exceed  c.  The  three  parts  are  denoted  here  by  y, 
x  —  y  and  a  —  x.  The  condition  that  x  —  y  shall  not  exceed 
c  cuts  off  from  the  whole  area  OAC,  as  we  have  just  seen,  the 
triangle  ADE.  The  condition  that  y  shall  not  exceed  c  cuts 
off  in  like  manner  the  triangle  above  the  horizontal  dotted 


D 

FIG.  47. 


§  XVIIL]     PROBABILITIES  REPRESENTED  BY  AREAS.        269 

line,  and  the  condition  that  a  —  x  shall  not  exceed  c  excludes 
that  to  the  left  of  the  vertical  dotted  line.  The  area  excluded 
is  in  each  case  \(a  —  cf\  but  when  (as  represented  in  Fig.  47) 
c  <  \a,  these  areas  overlap,  and  the  remaining  area  is  the 
small  isosceles  right  triangle  whose  area  is  readily  seen  to  be 
4(3^  —  of  (which  vanishes  when  c  =  \a,  as  should  evidently 

be  the  case).      Thus  the  probability  is  -   — 5 — -  when  c  is  be- 

ct 

/  \  rt 

tween  \a  and  \a\   and  it  is  I  —  3 ^ when  c  >  \a. 


Local  Probability. 

244.  Probability  questions  concerned  with  the  random  fall 
of  actual  points,  lines  and  other  geometrical  magnitudes  are 
sometimes  called  problems  of  local  probability.  One  of  those 
earliest  solved  was  proposed  by  Buffon  in  1777  as  follows:  A 
floor  of  indefinite  extent  is  ruled  with  equidistant  parallel  lines 
whose  common  distance  is  a ;  a  rod  of  length  c  is  thrown  upon 
it  at  random ;  what  is  the  chance  that  it  crosses  one  of  the 
lines? 

It  is  plain  that  one  end  of  the  rod  may  be  assumed  to  fall 
upon  a  fixed  line  perpendicular  to  the  parallel  lines.  Denot- 
ing by  x  its  distance  from  one  of  the  intersections  taken  as 
origin,  and  by  6  the  inclination  of  the  rod  to  this  line,  as  in 
Fig.  48,  all  values  of  x  and  of  6  are  equally  probable.  But 
considerations  of  symmetry  show  that  we  need  consider  only 
values  of  x  between  o  and  <z,  and  values  of  6  between  o  and 
^TI.  Hence  we  may  take  dx  dd  as  the  elementary  case,  and 
the  whole  number  of  cases  is 


IT 

rr 

^  o  J< 


dxdB  —    — . 
2 


2/O 


MEAN    VALUES  AND    PROBABILITIES.     [Art.  245. 


24-5.  In  counting  the  favorable  cases,  we  might  begin  by 
assuming  a  value  of  x,  taken  at  random,  to  be  fixed;  and,  after 

determining  the  number  of  favorable 
cases  consistent  with  this  value  of  xy 
sum  up  these  results  for  all  possible 
values  of  x.     Or  we  might  choose  0 
as  the    variable    to    be  regarded   at 
first  as  fixed.     In  making  this  choice, 
- — -•  we  are  in  fact  deciding  on  the  order 
of  integration.      In  this  case,  the  re- 
FlG-  48-  suit  is  simpler  if  we  regard  9  at  first 

as  fixed.  Now,  selecting  a  value  of  6,  we  see  that  favorable 
cases  occur  when  x  is  between  o  and  c  cos  (9,  the  limiting  values 
represented  in  the  lower  part  of  Fig.  48 ;  and  if  c  <  a,  this 
upper  limit  is  never  greater  than  a,  the  upper  limit  used  in 
finding  the  whole  number  of  cases.  Hence,  when  c  <  a,  the 
number  of  favorable  cases  is 


f-  fC  cos  g  ( 

dxdO  = 

Jo  Jo  J 


cos 


=  c\ 


and,  dividing  by  the  whole  number  of  cases,  the  probability  is 


-. 

na 


But  if  c  >  a  there  will  be  a  value  of  6,  as  shown  in  the 
upper  part  of  the  figure,  such  that,  for  any  smaller  value  of 
6,  all  values  of  x  give  favorable  cases.  The  integration  for 
6  must  now  be  separated  into  two  parts  at  this  point,  say 


6,  =  cos"1-.     The  number  of  favorable  cases  is  now 


XVIII.]  LOCAL   PROBABILITY.  2/1 


n 


,-    r 

dxdd-\- 

Jej 


—   re  cos  0 


—  adl  +  c(i  —  sin  ^)  =  a  cos-'  -  -j-  ^  — 

c 


which  gives 


The  limiting  value  of  this  expression  when  c  increases  without 
limit  is  of  course  unity. 

24-6.  When  a  problem  of  local  probability  involves  a  line 
drawn  from  a  given  point  in  a  random  direction  in  space,  it  is 
necessary  to  assume,  as  in  Art.  199,  a  spherical  surface  having 
the  given  point  for  centre,  and  to  regard  the  line  as  piercing 
this  surface  in  a  point  taken  at  random  upon  it.  For  example, 
a  shot  is  fired  with  a  given  velocity  in  a  random  direction  from 
a  point  on  the  circumference  of  a  circular  field  of  which  the 
diameter  2  a  is  equal  to  the  maximum  horizontal  range.  Given 
the  formula 

R  =  2a  sin  20     ......      (i) 

for  the  horizontal  range,  or  distance  at  which  the  shot  falls 
when  6  is  the  angle  of  elevation,  it  is  required  to  determine 
the  chance  that  the  shot  may  fall  within  the  field. 

Let  O,  Fig.  49,  be  the  point  of  projection,  and  let  the 
vertical  plane  through  the  line  of  fire 
make  the  angle  0  with  the  diameter 
OA  of  the  field.  Then  0  and  0  are  the 
spherical  coordinates  of  the  random 
point  on  the  surface  of  a  sphere  ol 
arbitrary  radius  b.  The  element  of 
surface  is 

d*S  =  P  cos  0  d<t>  dO,  FIG.  49- 

and  the  whole  surface,  representing  all  possible  cases,  is  that 


2/2  MEAN    VALUES  AND    PROBABILITIES-     [Art.  246. 

of  the  upper  hemisphere  27tb*.  Of  this  area  one-half,  ?r^2,  cor- 
responding to  values  of  0  on  the  second  and  third  quadrants,  is 
unfavorable.  A  portion  of  the  remaining  half  is  also  un- 
favorable and  we  proceed  to  find  its  value. 

For  a  value  of  0  between  •-  ^n  and  £?r,  as  in  the  dia- 
gram, those  values  of  6  are  unfavorable  which  make  R  >  r, 
the  radius  vector  of  the  field,  which  is 

r  =  2a  cos  0  .......      (2) 

Comparing  with  equation  (i),  we  have  therefore  an  unfavor- 
able case  if  sin  26  >  cos  0.  Now,  sin  2#  =  cos  0  when 


and  between  these  limits  (which  include  45°,  the  elevation  for 
maximum  range)  sin  20  >  cos  0,  hence  R  >  r.  Therefore  the 
additional  unfavorable  area  is 


/• 

=  24/2£2  r  sin  ^0^/0  =  44/2^2cos^0  k  —  4  (4/2  —  i)^2. 

Therefore  the  total  unfavorable  area  is  b\n  -\-  4(^2  —  i)],  and 
the  favorable  area  is 

b\7t  —  4(4/2  -  i)]. 
Thus  the  chance  required  is -,  which  is  about  .24. 

2  71 


XVIII.]         THE   ELEMENT   OF  PROBABILITY.  273 


The  Element  of  Probability. 

24-7.  If  the  element  which,  when  integrated  between  differ- 
ent limits,  gives  the  whole  number  of  cases  and  also  the  num- 
ber of  favorable  cases,  be  first  divided  by  the  whole  number  of 
cases,  the  result  is  an  element  of  probability.  If  this  element 
be  integrated  with  the  limits  proper  to  the  whole  number,  it  of 
course  gives  unity;  and  if  integrated  with  the  limits  proper  to 
the  favorable  number,  it  gives  the  probability  at  once.  Thus, 
when  every  value  of  x  between  o  and  a  is  equally  probable, 
the  element  of  probability  is  dx/a.  We  cannot  speak  of  this 
as  the  probability  of  a  special  value  of  x ;  it  may  however  be 
called  the  probability  of  a  value  between  x  and  x  -f-  dx.  In 
the  case  of  the  random  fall  of  the  point  P,  it  is  the  chance  that 
P  falls  upon  the  element  dx  of  the  line  a. 

24-8.  Now  suppose  that  a  point  P  falls  at  random  within 
a  circle  of  radius  a,  and  let  r  denote  its  distance  from  the  centre. 
All  values  of  r  between  o  and  a  are  now  possible,  but  it  is 
plain  that  they  are  not  equally  probable.  The  chance  that  r 
falls  between  r  and  r  -f-  dr  is  now  the  chance  that  P  falls  upon 
the  elementary  annulus  2nrdr;  therefore,  dividing  by  the  area 
of  the  circle,  it  is 

2.rdr 


Hence  the  probability  in  question  is  proportional,  not  to  dr, 
but  to  r  dr-*  that  is,  it  is  of  the  form  krdr.  But  the  value  of  k 
depends  upon  the  extreme  values  between  which  r  is  known 
to  fall,  and  may  be  determined  by  the  condition  that  the  value 

*  In  using  polar  coordinates,  the  probability  of  r  for  a  point  falling  at  random 
is  proportional  to  r  dr  when  6  has  a  fixed  value,  and  that  of  6  is  d$  when  r  has  a 
fixed  value.  Accordingly  the  probability  of  the  joint  occurrence  of  a  particular 
value  of  r  and  a  particular  value  of  6  is  proportional  to  the  product  r  dr  dQ. 


274  MEAN  VALUES  AND  PROBABILITIES.   [Art.  248. 

of  the  integral  between  these  limits  must  be  unity.  Thus,  if 
values  of  r  between  rl  and  rz  only  are  possible,  assuming  the 
form  fcrdrand  determining  k,  we  find,  for  the  element  of  prob- 
ability, 


In  like  manner,  if  a  point  falls  at  random  within  a  spherical 
surface  of  radius  a,  its  distance  from  the  centre  has  the  ele- 
mentary probability 

^r2  dr 

24-9.  When  the  element  of  probability  of  one  of  the  vari- 
ables involved  in  a  problem  is  independent  of  the  other  vari- 
ables, and  can  be  written  down  beforehand,  the  problem  can 
be  made  to  depend  upon  the  simpler  case  in  which  the  variable 
in  question  has  a  fixed  value.  Denote  this  variable  by  r,  and 
let  dp  be  its  element  of  probability,  which  we  suppose  a  known 
function  of  r  and  dr.  Let  P  denote  the  required  probability, 
and  P0  the  probability  when  the  value  of  r  is  fixed.  We  sup- 
pose P0  to  be  first  determined;  it  will  of  course  be  a  function 
of  r.  Then  the  product  P0dp  will  express  the  probability  that 
the  event  will  happen  in  connection  with  a  value  of  r  between 
r  and  r  -f-  dr.  But  this  is  only  one  way  in  which  the  event 
may  happen,  for  it  may  happen  in  connection  with  any  one  of 
the  possible  values  of  r.  Hence  the  entire  probability  of  the 
event  is  found  by  summing  up  the  probabilities  of  the  different 
ways,  that  is  by  integration.  Thus 

/?  = 

taken  between  the  limits  given  for  r.      The  method  is  analo- 
gous to  that  given  for  mean  values  in  Art.  222. 


XV II I.]        THE  ELEMENT  OF  PROBABILITY. 


275 


FIG.  50. 


250.  As  an  illustration  let  us  find  the  probability  that  the 
distance  between  two  points  A  and  B  taken  at  random  within 
a  sphere  shall  be  less  than  the  radius  a. 
The  probability  will  evidently  be  unaltered 
if  we  assume  one  of  the  points  A  to  be 
taken  on  a  fixed  radius  CD  of  the  sphere. 
Let  r  denote  its  distance  CA  from  the  centre. 
The  value  of  r  is  independent  of  the  other 
variables  (which  concern  the  position  of  the 
other  point,  B)  and  its  elementary  prob- 
ability is,  as  mentioned  in  Art.  248, 

3r2  dr 
*=—!-. 

Let  us  now  find  P0,  the  probability  that  AB  <  a  when  A  has 
the  position  given  in  Fig.  50. 

Let  a  spherical  surface  whose  centre  is  A  and  radius  a  be 
described,  then  P0  is  the  chance  that  B  shall  fall  in  the  lens- 
shaped  volume  cut  from  the  given  sphere  by  this  spherical  sur- 
face. In  other  words,  it  is  the  ratio  of  the  volume  of  the  lens 
to  that  of  the  sphere.  The  lens  is  double  the  segment  of  the 
sphere  cut  off  by  a  plane  perpendicular  to  and  bisecting  CA  at 
D.  Thus  its  volume  is 


2/7 


Pa 

(of  —  s?)dx  = 

hr 


and,  dividing  by  the  volume  of  the  sphere,  j^rra3,  we  have 

1 2  a~r  —  r3 


Therefore 


dP=P0dp  =     ~ -5 

0  ^  3  b 


and,  integrating  between  o  and  a, 

p  —  T  _  .    a/, i\  —  T 11  —  15 

TffU  f/  ~  ~S~S  —  ??' 

Thus  the  odds  are  17:15  in  favor  of  a  distance  greater  than  a. 


276  MEAN    VALUES   AND    PROBABILITIES.     [Art.  251. 


Curves  of  Probability. 

251.  The  elementary  probability  of  a  variable  x  of  which 
the  possible  values  are  not  equally  probable  may  be  put  in  the 
form  y  dx,  where  y  is  a  function  of  x.     The  curve  in  which  y  is 
the  rectangular  ordinate  corresponding  to  the  abscissa  x  (or 
rather  that  part  of  it  corresponding  to  possible  values  of  x~]  is 
called  the  curve  of  probability  for  the  variable  x.      Such  a  curve 
is  a  graphic  representation  of  the  law  of  probability  of  x,  as  it 
may  be  called,  that  is  the  mode  in  which  the  probability  of  x 
varies.     The  ratio  of  any  two  values   of  y  gives  the  relative 
probability  of  the  corresponding  values  of  x,  although  we  can- 
not assign  actual  values  to  the  probabilities  without  introducing 
the  element  dx.     Since  the  whole  probability  that  x  shall  fall 
between  given  limits  is  a  value  of  the  integral 

\ydx% 

the  probability  that  x  shall  fall  between  any  given  limits  is  rep- 
resented by  the  area  enclosed  between  the  curve,  the  axis  of 
x  and  the  ordinates  of  the  limiting  values.  For  this  purpose, 
the  total  area,  of  which  the  base  represents  the  whole  range  of 
possible  values  of  x,  must  of  course  be  taken  as  unity. 

252.  In  illustration,  let  us  find  the  law  of  probability,  and 
construct  the  probability  curve,  for  the  distance  from  one  end 

of  a  line  AB  =  a  of  the  nearer  of 
two  random  points  which  have 
fallen  upon  AB.  Let  P  and  Q  be 
the  two  points  which  fall  at  ran- 


B     com;  let  ^  denote  the  one  nearer 

FIG.  51.  to  A ,  and  x  the  distance  AX.      The 

chance  that  P  shall  fall  upon  any  given  element  dx  is  dx/a. 

If  this  happens,  the  chance  that  P  is  the  point  X  is  the  same 


§  XVI 1 1.]  CURVES   OF  PROBABILITY.  277 

thing  as  the  chance  that  Q  falls  on  the  segment  a  —  x  to  the 
right  of  dx.  Thus  the  chance  that  P  falls  on  dx,  and  is  the 
point  X,  is 

(a  —  x}dx 
~^        ' 

But  there  is  an  equal  chance  that  Q  falls  upon  dx  and  is  the 
point  X.  Therefore  the  whole  chance  that  X  falls  upon  dx,  or 
value  of  ydx,  is  in  this  case 

2  (a  —  x]dx 

-^r-' w 

hence  the  probability  curve  for  the  nearer  to  A  of  two  random 
points  is  the  oblique  straight  line 

2(a  —  x} 

**•**-?-*' 

which  passes  through  B  as  represented  in  Fig.  51.  Thus  the 
law  of  probability,  in  this  example,  is  a  uniform  decrease  of 
probability  from  a  maximum  at  A  to  zero  at  B.  The  ordinate 
A  C  corresponding  to  x  =  o  is  2/a ;  this  makes  the  whole  area 
of  the  triangle  ABC  unity,  as  it  should  be. 

253.  The  probability  that  x  shall  fall  between  any  given 
values  is  now  represented  by  a  part  of  the  area  of  this  triangle. 
Thus  the  probability  that  it  shall  be  less  than  AE  in  Fig.  51 
is  the  area  of  the  trapezoid  'AEFC .  It  is  therefore  the  same 
thing  as  the  probability  that  a  point  falling  at  random  upon 
the  triangle  ABC  shall  fall  upon  this  trapezoid.  So  also,  in  gen- 
eral, the  probability  curve  defines  an  area  such  that  the  variable 
for  which  it  is  constructed  may  be  regarded  as  the  abscissa  of 
a  point  falling  at  random  upon  the  area. 

In  particular,  if  we  draw  an  ordinate  which  divides  the  area 
into  two  equal  parts,  the  corresponding  abscissa  is  that  value 
which  x  is  just  as  likely  to  exceed  as  to  fall  short  of.  This 


278  MEAN    VALUES  AND    PROBABILITIES      [Art.  253. 

value  is  often  called  the  probable  value.  It  is,  of  course,  not 
generally  the  most  probable  value.  In  this  case,  in  fact,  any 
smaller  value  has  a  greater  relative  probability. 

254-.  In  like  manner,  the  dotted  line  in  Fig.  5 1  is  the  prob- 
ability curve  for  the  distance  of  the  farther  from  A  of  the  two 
random  points.  Its  equation  is 

2x  2x  dx 

y=^     hence     — p- 

is  the  element  of  probability.  That  is,  the  probability  of  the 
distance  x  of  the  more  distant  of  two  points  selected  at  random 
is  proportional  to  x  dx.  The  probable  value,  in  this  case,  is 
a  |/^ ;  that  is  to  say,  it  is  an  even  wager  that  the  greater  of  the 
distances  of  the  two  points  shall  exceed  this  value.  The 
alternative  event,  in  this  , case,  is  that  both  distances  shall  fall 
short  of  a  ^/£,  the  probability  of  which  is  also  equal  to  £. 

205.  We  have  seen  in  Art.  223  that,  when  two  random 
points  fall  upon  the  line  a,  the  distance  x  of  the  nearer  point 
from  one  end  is  one  of  three  random  parts  of  the  line ;  hence  the 
•expression  found  in  Art.  252  and  the  line  CB  in  Fig.  51  ex- 
press the  law  of  probability  of  one  of  the  parts  when  a  is 
divided  at  random  into  three  parts.  By  inspection  of  the  figure, 
it  is  evident  that  the  chance  that  the  part  shall  be  less  than  \a 
is  |;  the  chance  that  it  shall  exceed  %a  (its  mean  value)  is  -|; 
and  so  on. 

256,  If  three  points  fall  at  random  upon  the  line  AB,  and 
Ji^  Y,  Z  denote  them  when  selected  in  order  of  nearness  to 
A,  the  probability  curves  for  their  distances  from  A  will  be 
found  to  be 

_  3(«  —  *?  _  6Q  —  x)x  _  3^ 

y  ~~         ~a3         '    y  ~~  a3         '    y  :  :   a3  ' 

The  proof  is  similar  to  that  in  Art.  252.    Thus,  for  the  middle 


XVIII.] 


CURVES   OF  PROBABILITY. 


279 


point:  The  probability  that  the  points  fall  in  the  order  P,  Q,  R, 
and  that  Q  falls  on  a  given  element  dx\  is  the  probability  of 
the  joint  event  that  Q  falls  on  dx,  R  falls  on  the  segment 
a  —  x  to  the  right  of  it,  and  P  on  the  segment  x  to  the  left 
of  it.  The  respective  probabilities  of  these  events  are 


dx 

a 


a  —  x 


and 

a 


Hence  the  probability  of  the  compound  event  is 
(a  —  x}x  dx 


But  this  is  only  one  way  in  which  the  middle  point  can  fall  on 
dx,  for  there  are  six  orders  in  which  P,  Q  and  R  may  fall ; 
hence  the  whole  probability  that  X  falls  upon  dx  is 

6(a  —  x]x  dx 


Taking  A  as  before  for  the  origin,  the  three  curves  are  the 
parabolas  shown  in  Fig.  52.  The  first,  CB,  for  the  point 
nearest  to  A ,  is  also  the  proba- ; 
bility  curve  for  the  value  of 
one  of  four  parts  into  which  a 
is  divided  at  random.  It  will 
be  noticed  that  the  sum  of  the 
three  elementary  probabilities  FIG.  52. 

is  - — ;  as  should  be  expected,  since  the  sum  of  the  chances 
a 

that  X,  Y  and  Z,  respectively,  shall  fall  upon  a  given  dx  is 
evidently  the  same  as  the  sum  of  the  chances  that  P,  Q  and 
R  shall  fall  upon  dx. 


2 SO  MEAN    VALUES  AND    PROBABILITIES-     [Art.  257. 


Discontinuous  Curves  of  Probability. 

257.  Since  the  ordinates  of  a  probability  curve  are  merely 
graphic  representations  of  the  relative  probabilities  of  the  cor- 
responding values  of  x,  any  curve  in  which  the  ordinates  are 
proportional   to  these  relative  probabilities    will  serve  as  the 
curve  of  probability.      Then,   as  mentioned    in  Art.  253,  the 
equally  probable  cases  correspond  to  the  falling  at  random  of 
a  point  upon  equal  elementary  areas  of  the  probability  curve. 
Thus  the  area  whose  base  is  a  given  range  of  values  represents 
the  number  of  favorable  cases;  and,  to  obtain  the  numerical 
value  of  the  probability,  this  must  be  divided  by  the  whole 
area  which  represents  the  whole  number  of  cases,  or,  as  we 
may  say,  the  probability  unity. 

258.  The  probabilities  of  the  values  of  the  variable  x  will 
frequently,  as  a  result  of  given  conditions,  follow  different  laws 
in  different  parts  of  the  range  of  its  possible  values ;  in  other 
words,  the  probability  will  be  a  discontinuous  function  of  x, 

When  the  variable  is  represented  by  one  coordinate  of  a 
point  which  falls  at  random  upon  an  area  defined  by  limits  of 
integration,  this  area  will  at  once  determine  the  law  of  probabil- 
ity for  all  values.  For  example,'  in  Art.  226  we  saw  that  the 
restrictions  upon  x  and  y,  which  are  necessary  to  make  x 
represent  the  least,  and  a  —  y  the  greatest,  of  three  random 
parts  into  which  a  is  divided,  limit  the  point  (x,  y)  to  the 
triangular  area  OEF  in  Fig.  43.  Thus  the  least  part  is  the 
abscissa  of  a  point  falling  at  random  upon  this  triangle.  In- 
spection of  the  diagram  shows  that  no  value  of  x  greater  than 
\a  is  possible.  If  a  line  parallel  to  the  axis  of  y  be  drawn 
corresponding  to  any  smaller  value  of  x,  the  segment  of  it 
included  within  the  triangle  measures  the  relative  probability 
of  that  value  of  x.  Thus  the  diagram  shows  that  the  prob- 
ability of  a  given  value  of  the  least  part  decreases  uniformly 


XVIIL]  FACILITY   OF  ERRORS   OF  OBSERVATION.  28 1 


from  a  maximum  at  the  value  zero  to  nothing  at  the  value  \a. 
No  discontinuity  occurs,  in  this  case,  between  the  extreme 
possible  values. 

259.  Now  consider  the  greatest  part,  which  was  represented 
by  a  —  y.      This  is  the  distance  of  the  point  (x,  y]  from  the  lino 
BC,     The  diagram  shows  that  the  possible  values  lie  between  \  r 
and  a.    The  relative  probability  of  a  given  value  is  measured  by 
the  segment  within  the  triangle  of  a  line  parallel  to  the  axis  of  .rat 
the  given  distance  from  BC.    The  maximum  probability  occurs, 
therefore,  at  the  value  -^a.     From  this  maximum  the  probability 
decreases   uniformly  to   zero   at  the  value   a,  and  it  also  de- 
creases uniformly  to  zero  as  we  pass  from  \a  to  \a.      Thus  the 
curve  of  probability,  when  laid  down 
upon  the  axis  of  4r,  is  the  broken  line 
NPA  in  Fig.  53.      From  this  figure  it 
is  readily  shown  that  the  probability  _ 
that  the  greatest  of  three  parts  exceeds   * 
\a  is  f ,  the  probability  that  it  exceeds 
#  is       ,  and  so  on. 


N       M 


53- 


Law  of  Facility  of  Errors  of  Observation. 

260.   In  the  Theory  of  Errors   of  Observation,  the   prob- 
ability of  the  occurrence  of  an  error  x  is  assumed  to  be  propor- 
tional to  e~;''2-**,  where  h  is  a  con- 
stant   depending  upon  the  pre- 
cision of  the  observations.      The 
law  implies  (as  indicated  by  the 
form    of  the    curve  y  =  ce-'1"1*'* 
shown  in  Fig.  54)  that  positive 
and  negative  errors  numerically 
•^    equal  have  the  same  probability, 
FlG«  54-  that    the  maximum    probability 

occurs  at  x  =O,  and  that  it  becomes  so  small  for  large  values 


282  MEAN    VALUES  AND    PROBABILITIES-     [Art.  260. 

of  x  that  it  is  practically  unnecessary  to" assign  any  finite  limits 
to  the  possible  errors.  Since  the  number  of  errors  between  x 
and  x  -)-  dx  which  may  be  expected  in  ' '  the  long  run  ' '  is 
proportional  to  ce~h***dx,  the  curve  is  often  said  to  express  t/te 
law  of  frequency  of  errors;  or  the  law  of  facility  of  the  error  x. 
In  order  that  ce~k*x*dx  shall  be  equal,  and  not  simply  pro- 
portional, to  the  probability,  it  is  necessary  that  the  whole  area 
of  the  curve  shall  be  unity,  or 


which  can  be  shown  to  give  c-=— — .    See  Art.  280. 

I/7T 


Mean  Values  under  Given  Laws  of  Probability. 

261.  In  equation  (2),  Art.  189,  pl  ,  p.2  etc.  express  the 
relative  frequencies  with  which  the  values  zl  ,  zz  etc.  of  a  vari- 
able z  occur  among  the  values  of  which  the  mean  is  required. 
Dividing  by  2p,  the  whole  number  of  cases,  the  equation 
becomes 

' 


in  which  the  coefficients  of  the  several  values  are  their  prob- 
abilities.     Denoting  these  by  P^,  P2  etc.,  we  have 


M=   P^  +  P,Z,  +...==    ?Pz.      ...         (I) 

Thus  the  mean  value  under  a  given  law  of  probability  may  be 
defined  as  the  sum  of  the  products  of  the  several  values  each 
multiplied  by  its  own  probability. 

In  finding  the  mean  of  a  continuous  variable  z,  the  element 
which  takes  the  place  of/,  when  integration  takes  the  place  of 
summation,  expresses  the  relative  frequency  or  probability  of 


§  XVIII.]       MEAN    VALUES    UNDER    GIVEN   LAWS.  283 

the  various  values  of  z,  in  the  distribution  for  which  M  is 
required.  Denoting  this  element  by  dp,  the  formula  for  the 
mean  is 

'    dp (2) 


in  which  we  may  regard  dp  as  an  element  proportional  to  the 
probability  of  z.  But,  when  the  actual  element  of  probability 

dP  is  employed  (so  that  dP  =  i  when  the  integration  corre- 
sponds to  the  whole  range  of  values  of  z  for  which  M  is  re- 
quired), the  formula  becomes 

M=  \zdP (3) 


262.  For  example,  in  the  theory  of  errors  the  mean  error, 
denoted  by  e,  is  denned  as  that  whose  square  is  equal  to  the 
mean  square  of  an  error.  Hence,  given  the  law  of  frequency, 
dp  =  e~/'^'i  dx,  we  have 


Integrating  by  parts,  we  have 


in  which  the  first  term  vanishes  at  each  limit.      Hence 

i  i 

e   —  T75 ,     or     e  = 


kl/2' 

Again,  the  mean  value  of  the  error  without  regard  to  sign  is, 
in  the  same  theory,  denoted  by  ?;;   hence,  using  the  exact  ele- 

ment of  probability,  which  (see  Art.  260)  is  dP—  —e-*wdx 

tfn 

we  have 

°°  i 

xe-»**dx  =  -  ~-e- 


h    f° 
ff  =  2— 

V*J0 


284  MEAN   VALUES  AND   PROBABILITIES.     [Art.  262. 

It  will  be  noticed  that  e  is  the  radius  of  gyration  of  the  area 
in  Fig.  54  about  the  axis  of  y,  and  77  is  the  abscissa  of  the 
centre  of  gravity  of  the  area  on  the  right  of  that  axis. 

Probabilities  Involving  Selected  Points. 

263.  In  any  question  of  probabilities  involving  a  variable 
x  whose  values  are  not  equally  probable,  but  follow  the  same 
known  law  throughout  its  possible  values,  the  element  express- 
ing this  law  takes  the  place  of  the  simple  element  dx.  Thus, 
suppose  that,  in  the  problem  of  Art.  241,  Y  instead  of  being 
taken  at  random  is  the  farthest  from  O  of  three  points  taken  at 
random,  and  that  X  is  the  farther  from  0  of  two  other  points 
taken  at  random  ;  required  as  before  the  chance  that  the  dis- 
tance YX  shall  exceed  c. 

Assuming  that  y  <  x,  as  in  Fig.  46,  we  proceed  as  in  Art. 
241,  except  that,  in  accordance  with  Arts.  254  and  256,  dx  is 
replaced  by  x  dx,  and  dy  by  y*  dy.  Thus  the  whole  number  of 
cases  is  now. 

fa  a5 

yz  dy  xdx  =  \  \  x4  dx  =  —  ; 

J  o  15 

and,  for  the  number  of  favorable  cases,  we  have 

n*-c  fa 

y*  dyxdx  =  f     (x  —  cfx  dx. 
*     o  '  c 

Putting,   for  convenience,   in  this   last   integral   z  =  x  —  c,  it 
becomes 

j  r*.  +  cy,  = 

Jo 


20  J 


Hence  the  probability  is  P.  =  (*  -  c)4(4*  + 

J 


§  XVIIL]     PROBABILITIES   OF    SELECTED    VALUES.          285 

But  if  y  >  x,  the  probability  takes  a  different  form.      The 
whole  number  of  cases  is  now 


Jy  fa  a5 

x  dx  y*  dy  =  k      j/4  dy  =  — , 


and  the  favorable  number  is 

fa  ey-c 


17 

J  cJ  o 


xdxfdy  = 


/  \s 

hence  the  probability  is  Pz  =  --.,  5      (6a2  -j-  "$ac  -(-  £2). 

264-.  If  we  require  the  probability  that  the  distance  shall 
exceed  c,  without  distinction  of  the  cases  whenj  <  x  and  y  >  x, 
we  must  compare  the  sum  of  the  favorable  cases  with  the  total 
numbers.  The  sum  of  the  latter  is  ^a5,  which  is  in  fact  the 
value  of 


IT 

J  oJ  r 


y*  dy  x  dx. 


The  sum  of  the  favorable  cases  will  be  found  to  reduce  to 

(a  — 


Hence  the  probability  that  the  distance  without  regard  to  sign 
shall  exceed  c  is 


a-' 


265.  If,  in  the  problem  solved  above,  we  represent  x  and 
y  by  the  rectangular  coordinates  of  a  point,  as  in  Art.  242,  the 
point  (x,  y]  is  restricted  to  the  square  OA  CB,  Fig.  5  5 .  The 
total  number  of  cases  represented  by  the  integral  of  the  pre- 


286 


MEAN    VALUES  AND    PROBABILITIES.    [Art.  265. 


ceding  article  may  now   be   regarded   as  a   large   number  of 
points  distributed  not  uniformly  over  the  area,  but  in  such  a 

^ G_       G     way  that  the   number  falling  upon  any 

element  of  surface  dy  dx  is y^xdydx.  In 
other  words,  the  points  are  distributed 
with  a  density  proportional  to  the  value 
of  y^x.  The  point  (x,  y}  must  now  be 
considered  as  one  taken  at  random  from 
this  large  number  of  points ;  and  the 
probability  that  it  comes  from  a  certain 
1'iti-  55-  favorable  area,  determined  by  the  limits 

of  integration,  is  the  ratio  of  the  number  of  points  in  the  favor- 
able area  to  the  whole  number. 

Thus,  in  the  present  problem,  if  lines  DE  and  FG  parallel 
to  the  diagonal  OC  be  drawn  cutting  off  OD  and  OF  each 
equal  to  c,  the  favorable  area  when  y  <  x  is  the  triangle  DAE. 
Hence  the  probability  P^  found  in  Art.  263,  is  the  ratio  of  the 
number  of  points  in  DAE  to  that  in  OAC.  In  like  manner, 
P2  is  the  ratio  of  the  number  in  FBG  to  that  in  OBC,  and 
finally  P  is  the  ratio  of  the  number  in  the  two  triangles  to  that 
in  the  entire  square. 

Selected  Points  upon  an  Area. 

266.  We  have  seen,  in  Art.  254,  that  the  selection  of  the 
more  distant  from  a  fixed  point  of  two  points  taken  at  random 
upon  a  straight  line  is  equivalent  to  giving  it  a  probability  pro- 
portional to  the  distance.  Suppose  now  that  two  points  fall  at 
random  upon  a  circular  area  of  radius  a,  and  that  we  select  the 
more  distant  from  the  centre ;  let  us  find  the  probability  that 
it  falls  upon  a  given  element  of  area. 

Denoting  the  random  points  by  P  and  Q,  and  by  X  that 
which  is  farther  from  the  center  O,  the  probability  that  P  falls 


§  XVIII.J     SELECTED   POINTS    UPON  AN  AREA.  287 

dS 

upon  a  given  element  dS  of  surface  is  —  -y  and  the  probability 

71  Ct 

that  P  is  X  is  then  the  probability  that  Q  falls  on  the  area 
nearer  to  O  than  dS>  which  bears  to  the 

r2 
whole  circle  the  ratio  —  .    Hence,  doubling 

CL 

the  product,  because  Q  may  fall  upon  dS 
and  be  X,  the  whole  probability  that  X 
falls  on  dS  is 


rta*  no  FIG.  56. 

267.  To  find  the  probability  that  X  falls  upon  any  given 
area  traced  out  upon  the  circle,  we  have  only  to  integrate  this 
over  the  given  area.  For  example,  if  the  given  area  is  the 
circle  r  =  a  cos  0  in  the  diagram,  having  for  diameter  one  of 
the  radii  of  the  large  circle,  we  have 


4    r-    f 
=- 

7ta*)0    J0 


f*  cos  e  -1  - 

P=-  r*drde  =  -  V   cos4  8  d6  =  4- 

16 


The  odds  are  therefore  13:  3  that  the  more  distant  of  the  two 
points  shall  fall  outside  of  the  small  circle. 

Random  Lines. 

268.  A  straight  line  is  said  to  be  drawn  at  random  in  a  plane 
if  all  directions  are  equally  probable,  while,  among  the  lines 
having  a  given  direction,  all  points  of  intersection  with  a  common 

*  The  points  Jfare  here  distributed  with  a  density  proportional  to  r2,  but  the 
values  of  r  have  a  probability  proportional  to  r3.  The  result  would  apply  to  any 
area  upon  which  points  had  this  distribution,  but  it  is  to  be  noticed  that  selection 
of  the  more  distant  from  a  fixed  point  of  two  points  upon  an  area  not  a  circle  would 
not  produce  this  distribution  except  within  a  circle  whose  radius  is  the  least  value 
of  r  upon  the  boundary  of  the  given  area. 


288  MEAN    VALUES  AND    PROBABILITIES.    [Art.  268. 

perpendicular  are  equally  probable.  In  expressing  the  whole 
number  of  lines  which  cut  a  given  convex  curve,  let  p  be  the 
perpendicular  let  fall  upon  the  line  from  some  fixed  point  within 
the  curve,  and  let  0  be  the  inclination  of  this  perpendicular  to 
some  fixed  direction.  Then  the  whole  number  will  be  found 
by  the  integration  of  dp d<£>.  If,  in  this  integration,  the  limits 
taken  for  p  are  zero  and  the  perpendicular  upon  a  tangent  to 
the  curve,  0  must  take  all  values  from  o  to  2n.  Thus  the 
number  of  lines  which  cut  the  curve  will  be 


r2"-  f/  r* 

dp  dtp  = 

J o   •  o  J 


where  /  is  the  perpendicular  upon  the  tangent  to  the  curve. 

269.  Now  it  is  shown  in  Diff.  Calc.,  Art.  348  (and  is  geo- 
metrically evident  on  drawing  the  figure)  that,  if  r  is  the  part 
of  the  tangent  intercepted  between  the  point  of  contact  and  the 
foot  of  the  perpendicular, 

df  =  ds  —  pd$t 

where  s  is  the  length  of  the  arc  measured  from  some  fixed 
point.  Now  when  we  have  completed  the  whole  circuit  of  the 
curve,  so  that  T  returns  to  its  original  value,  the  integral  of  dr 
is  zero ;  it  follows  that,  denoting  by  L  the  whole  length  of  the 
curve,  we  have 


L 


f2 
= 

J  o 


Therefore  L,  the  length  of  the  curve,  is  the  measure  of  the  num- 
ber of  lines  drawn  at  random  which  meet  the  curve.  Thus,  if 
a  convex  curve  of  length  /  is  drawn  within  a  convex  curve  of 
length  L,  the  chance  that  a  random  .line  which  cuts  L  shall  also 
cut  /  is  l/L. 


§  XVIII.]  RANDOM  LINES.  289 

270.  If  the  given  line  is  not  convex,  or  if  it  is  not  closed, 
the  same  result  holds  if  L  denotes  the  length  of  the  shortest 
convex  closed  line  which  surrounds  the  given  line  like  an  elas- 
tic band  stretched  about  it;  for  it  is  evident  that  a   straight 
line  which  cuts  the  given  line  must  cut  such  a  band. 

For  example,  the  number  of  lines  which  pass  between  two 
given  points  whose  distance  is  c  is  measured  by  ic.  Thus,  if 
two  points  on  the  circumference  of  a  circle  subtend  an  angle  a 
at  the  centre,  the  chance  that  a  random  line  cutting  the  circle 
shall  cut  both  of  the  arcs  into  which  it  is  divided  by  the  points 
is  (2  sin  \a)/7t. 

Probabilities  involving  Variable  Magnitudes. 

271.  When  the  probability  of  an  event  depends  upon  a  vari- 
able magnitude  in  such  a  way  that  the  probabilities  correspond- 
ing to  any  values  of  the  variable  are  proportional  to  the  values 
themselves,  the  actual  probability  of  the  event  is  the  same  as 
that  corresponding  to  the  mean  value  of  the  variable. 

For,  by  hypothesis,  the  probability  that  the  event  will  hap- 
pen when  z  has  the  value  z^  ,  z2  ,  .  .  .  and  M  are  respectively 

^      z2  M 

-,     —  ,  .  .  .       and        -  ; 
a       a  a 

where  a  is  a  constant. 

Now,  dividing  equation  (i),  Art.  261,  by  a,  we  have 


Since  Pl  is  the  probability  of  the  value  sl  ,  the  first  term  of  the 
second  member  is  the  chance  that  z  shall  have  the  value  zv 
and  that  the  event  shall  then  happen.  But  this  is  only  one 
way  in  which  the  event  can  happen,  and  the  second  member  is 
the  sum  of  the  probabilities  of  its  happening  in  all  possible 
ways,  that  is,  the  total  probability  of  the  event.  Hence  the 
equation  expresses  the  proposition  to  be  proved. 


MEAN   VALUES  AND  PROBABILITIES.     [Art.  272. 

272.  In  particular,  if  a  in  equation  (i)  is  a  fixed  area  and  2 
is  a  variable  area  within  it,  the  chance  that  a  point  following 
at  random  upon  a  shall  fall  upon  s  is  M/a,  where  Mis  the 
mean  value  of  the  area  z.  For  example,  let  A  be  the  area  of  a 
given  triangle  ABC,  and  z  that  of  the  triangle  PQR,  where 
P,  Q  and  R  are  three  points  taken  at  random  on  ABC;  then 
we  have  found  in  Art.  238  that  the  mean  value  of  PQR  is  -£%A . 
It  follows  that  if  three  points  be  taken  at  random  on  ABC,  the 
chance  that  a  fourth  point  taken  at  random  on  ABC  shall  fall 
within  PQR  is  j1^.  Hence,  also,  if  four  points  are  taken  at  ran- 
dom, the  chance  that  one  of  the  four  shall  fall  within  the  tri- 
angle found  by  the  other  three  is  |-.*  On  the  other  hand,  the 
chance  that  they  shall  form  the  vertices  of  a  convex  quadri- 
lateral is  f . 

Conversely,  the  mean  value  of  an  area  may  sometimes  be 
found  by  knowing  the  probability  that  a  point  falls  upon  it. 
Thus,  in  the  foregoing  illustration,  the  sides  of  the  triangle 
PQR  when  produced  separate  the  whole  triangle  ABC  into  six 
other  parts  beside  the  triangle  PQR.  If  S  is  a  fourth  point 
taken  at  random,  the  chance  that  P  falls  within  the  triangle 
QRS  is  the  same  as  the  chance  that  5  falls  within  PQR.  Hence 
the  mean  area  of  the  space  in  the  vertical  angle  of  the  angle  at 
P  is  also  -faA.  We  thus  have  four  areas  each  of  whose  mean 
values  is  -^A,  and  it  readily  follows  that  each  of  the  other 
three  has  the  mean  value  \A. 

273.  Since  the  chance  that  a  given  line  shall  be  cut  by  a 
random  line  chosen  from  a  given  set  of  random  lines  is  propor- 
tional to  its  length,  the  chance  of  cutting  a  variable  line  is  the 
same  as  the  chance  of  cutting  its  mean  value.  For  example, 


*  If  P,  Q,  R  and  5" are  the  points  "  /'falls  within  QRS,"  "^  falls  within  PJRS," 
etc.  are  mutually  exclusive  events;  that  is,  no  two  of  them  can  happen  at  once. 
The  probability  that  some  one  of  such  a  set  of  events  shall  happen  is  evidently  the 
sum  of  their  respective  probabilities. 


§  XVIII.]  VARIABLE   MAGNITUDES.  2QI 

a  random  line  is  drawn  cutting  a  circle,  and  two  points  are 
taken  at  random  on  the  circumference  ;  what  is  the  chance  that 
they  lie  upon  opposite  sides  of  the  line  ?  This  is  clearly  the 
same  thing  as  the  chance  that  a  random  line  shall  cut  the 
chord  joining  two  random  points.  The  mean  value  of  such 

a  chord  was  found  in  Art.  218  to  be  M  =  --  ;    hence  (see 

71 

Art.  270)  the  chance  required  is  2M/27ta  or  —  . 

274.  It  should  be  noticed  that  the  question  just  solved  is 
not  the  same  thing  as  the  chance  that  two  random  secants  shall 
cut  one  another  within  the  circle.  The  mean  value  of  the  por- 
tion of  such  a  secant  intercepted  by  the  circle  is  obviously  the 
double  of  the  mean  ordinate  to  a  given  diameter  which  was 
found  in  Art.  192  to  be  \na.  Therefore  M=  %rra,  and  the 

2M        I 

chance  in  this  case  is  -  =  —  . 

2 


The  mean  value  of  a  '  '  chord  '  '  in  the  sense  employed  above, 
that  is  when  the  line  or  secant  of  which  it  is  part  (not  the 
extremities  of  the  chord)  is  taken  at  random,  admits  of  a 
simple  expression  for  any  convex  curve.  For,  the  number  of 
such  chords  is 


JJ« 


\dpd<j>=.L\ 

and,  if  C  is  the  length  of  the  variable  chord, 
L.M=  \\Cdp  d$ 

determines  the  mean  value.  Now,  in  this  integral,  we  may 
take  as  limits  for  /  the  values  of  the  two  perpendiculars  cor- 
responding to  the  same  value  of  0  (one  of  which  will  be  nega- 
tive if  the  origin  is  taken  within  the  curve  as  in  Art.  268), 
provided  0  varies  only  between  the  limits  o  and  n.  Then  the 


MEAN    VALUES  AND  PROBABILITIES.  [Ex.  XVIII. 

value  of  the  integral  of  Cdp  will  be  A ,  the  area  of  the  convex 
curve,  independently  of  the-  value   of  0.      Hence  we  have 

TtA 


This  gives,  for  the  chance  that  two  such  random  chords  inter- 
sect, that  is  that  two  random  secants  intersect  within  the  area, 

_  2M  _  2nA 

p    ~T~    ~iy 

Examples  XVIII. 

i.  A  floor  is  ruled  with  parallel  lines  at  distances  20,  and  also 
with  another  set  at  distances  2b  perpendicular  to  them  ;  a  rod  of 
length  2C  less  than  either  of  the  distances  is  thrown  upon  the  floor 
at  random.  What  is  the  probability  that  it  crosses  a  line  ? 


2.  Three  points  are  taken  at  random  on  the  circumference  of  a 
"circle ;  what  is  the  chance  that  no  diameter1  can  be  drawn  having 
all  three  points  on  one  side  of  it  ?  £. 

3.  Show  that  the  chance  that  one  of  three  random  parts  of  a  is 
between  c,  and  a— c  is  the  same  as  that  for  one  of  two  random  parts. 

4.  What  is  the  chance  that  the  middle  point  of  three  random 
points  falling  on  a  line  shall  fall  on  the  middle  third  of  the  line  ? 

H- 

5.  Two  points  are  taken  at  random  in  the  northern  hemisphere; 
show  that  the  probability  that  their  difference  of  latitude  exceeds  a  is 

P  =  cos  a  —  (J  TT  —  «)sin  a. 

6.  Two  points  are  taken  at  random  upon  a  semicircle  and  their 
ordinates  drawn  ;  find  the  chance  that  a  point  taken  at  random  upon 
the  diameter  shall  fall  between  the  ordinates.  4  . 

~n* 

7.  Two  points  are  taken  at  random  within  a  circle  of  radius  a  ; 
find  the  probability  that  their  distance  shall  exceed  the  radius. 

3^3 


§  XVIII.]  EXAMPLES.  293 

8.  If  Fis  the  farther  from  O  of  two  points  taken  at  random 
upon  a  line  OA  of  length  a,  and  X  the  farther  of  two  other  points 
taken  at  random,  what  is  the  probability  that  YX  shall  exceed  c  ? 

(a  - 


g.  Supposing  X  in  the  preceding  problem  to  be  the  point  farther 
from  A  of  the  second  pair,  find  the  chance  that  the  distance  YX  shall 
exceed  c  :  first,  when  the  points  are  known  to  fall  in  the  order  O  YXA  ; 
second,  when  the  order  is  OXYA\  third,  when  the  order  is  not  known. 

(a  -  cY  .  (a-cY(a  +  <:-)(sa  +  c} 

~         '  ' 


3* 

10.  What  is  the  chance  that  the  distance  exceeds  c,  if  A"  and  Y 
are  the  farthest  from  O  respectively  of  two  and  of  four  points  ? 

P  =  t=A'(r5*'  -  3aV  +  6ai-  +  2,3). 

11.  Two  points  are  taken  at  random  in  the  northern  hemisphere; 
find  the  elementary  probability  of  the  smaller  of  their  latitudes. 

2(1  —  sin  (/))  cos  0  d(f>. 

12.  A  line  a  is  divided  into  three  parts   at  random,  and  that,  of 
intermediate  value  is  taken  ;  what   is   its  most    probable  value,  and 
what  the  "probable  value"  in  the  sense  explained  in  Art.  253  ? 

\a  ;  .289^. 

13.  What  is  the   probability  that  Y,  the  nearec  to  the  centre  of 
the  two  points  in  Art.  267,  falls  upon  the  given  area  ?        P  —  T5^. 

14.  A  point  is  taken  at  random  within  a  circle  whose  radius  is  a, 
and  a  line  is  drawn  at  random  through  it  ;  find   the   chance  that  it 
cuts  a  concentric  circle  whose  radius  is  c  <  a. 

c 

2  sin~  l—  . 

p_   _  a  +2f  ^(a1  —  S) 

TT  no1 

15.  If,  in  Ex.  14,  the  line  is  drawn  at  random,  what  is  the  proba- 
bility ?  c 

a 


294  MEAN    VALUES  AND    PROBABILITIES.  [Ex.  XVIII. 

16.  If  two  points  are  taken  at  random,  one  on  the  surface  and  the 
other  within  a  sphere  of  radius  r,  find  the  probability  that  their  dis- 
tance shall  be  less  than  c,  supposing  c  <  2r.  c'^r  —  3*:) 

i6r4 

17.  If  P  is  the  probability  that  the  distance  between  two  points 
taken  at  random  within  a  sphere   shall  be  less  than  c,  and  P0  that 
found  in  Ex.  16,  show  that  d(NP)  =  PQdN  (compare  Art.  222),  and 
thence  find  the  value  of  P.  _  S&r3  —  i&r'-f-  <:*)_ 

6 


18.  Find  the  mean  distance  of  the  middle  of  three  points  taken 
at  random  on  a  line  a  from  the  middle  of  the  line.  T\<z. 

19.  If  a  point  Z  is  taken   at  random   upon  a  line  OA  =  a,  and 
then  a  point  X  is  taken  at  random  on  OZ,  determine  the  probabil- 
ity curve  for  OX  =  x;  find  also  its   mean  value,  and  that   of  its 
square.  i  ,      a  a  a* 

y  =  a^x>          4;          9' 

20.  Lines  of  length  b  and  b'  fall  at  random    upon  a  line  a  ;    find 
the  chance   that  they  overlap  by  an  amount   less   than  c,  where  c  is 
less  than  either  b  or  b',  and  a  -j-  c  >  b  +  b'.         c(?a  —zb—2b'-\-c) 

(a  -*)(*-  *)  . 

21.  A  and  B  are  inhabitants  of  a  city  which  is  known  to  be  sit- 
uated on  a  river.     Assuming,  in  default  of  any  knowledge,  that  the 
river  divides  the  number  of  inhabitants  into  two  parts  at  random,  if 
it  is  known  that  B  lives  on  the  right  bank,  what  is  the  probability 
that  A  lives  on  that  bank?  |. 

22.  If,  in  the  preceding  example,  m  inhabitants  are  known  to  live 
on  one  side,  and  n  on  the  other,  show   that  the  odds  that  A  lives 
on  the  first-mentioned  side  are  m  +  i  :  n  +  i. 

23.  A  line  crosses  a  circle  at  random;    find  the  chance  that  a 
point  taken  at  random  within  the   circle  shall  be  distant  from  the 
line  by  more  than  the  radius  a  of  the  circle.  2 

3* 


§  XVIII.]  EXAMPLES.  295 

24.  Two  random  chords  of  a  circle,  that  is  chords  whose  extrem- 
ities are  taken  at  random,  are  drawn  ;   what  is  the  chance  that   they 
intersect  ?  \. 

25.  Two  random  lines  cut  a  square;  what  is  the  chance  that  they 
intersect  within  the  square  ?  n 

8" 

26.  A  line  is  drawn  at  random  across  a  circle;  what  is  the  chance 
that  two  points  taken  at  random  within  the  circle  shall  lie  on  oppo- 
site sides  of  it  ?  128 

45*" 

27.  Two  points  A   and  B  are  taken  at  random  in  a  triangle; 
what  is  the  chance  that  two  other  points  taken  at  random  in  the 
triangle  shall  fall  on  opposite  sides  of  AB  ?  ^. 

28.  Four  points  are  taken  at  random  within  a  circle;  what  is  the 
chance  that  they  form  the  vertices  of  a  convex  quadrilateral  ? 

35 

1271* 


DEFINITE  INTEGRALS.  [Art.  275 

CHAPTER   V. 

DEFINITE  INTEGRALS. 


XIX. 

Differentiation  of  a  Definite  Integral. 

275.  THE  general  symbol  for  a  definite  integral  of  a  single 
independent  variable  is 


f  f(x)dx] 

J  a 


where  a  and  b  are  constants,  that  is  to  say,  independent  of  x 
Denoting  its  value  by  u,  we  have  seen  in  Art.  82  that 


F(a),    .......     (i) 

where  F(x)  is  such  a  function  that 

dF(x] 


dx 


=/(*),         ........       (2) 


provided  F(x}  varies  continuously  while  x  passes  from  the  value 
a  to  the  value  b.  Moreover,  this  condition  will  be  fulfilled  if 
f(x)  is  itself  one- valued,  finite  and  continuous  for  the  same  range 
of  values  of  x.  The  independent  variable  x  is  used  only  in 
defining  the  integral  which,  by  equation  (i),  is  a  function  not 
of  x  but  of  the  limits.  It  may  be  called  the  current  -variable 


§  XIX.]   DIFFERENTIATION  OF  A    DEFINITE  INTEGRAL.     297 

when  other  variables  are  also  under  consideration,  and  it  is  evi- 
dently immaterial  what  symbol  is  used  for  the  current  variable. 

We  have,  in  Section  VII,  derived  the  values  of  certain  definite 
integrals  by  means  of  formulae  of  reduction  which  took  simple 
forms  by  virtue  of  special  values  of  the  limits.  We  shall  in  this 
chapter  consider  other  methcds  by  which  such  integrals  can  be 
evaluated  in  cases  where,  for  the  most  part,  the  value  of  the 
indefinite  integral,  F(x),  cannot  be  obtained. 

276.  Regarding  the  upper  limit  in 


r 
u  =     f(x)dx 

Ja 

as  variable,  we  have  from  equations  (i)  and  (2) 

and  in  like  manner  for  the  lower  limit 

—  =  -F'(a)=  -/(a) (2) 

If  the  limits  were  functions  of  some  other  variable  z,  we  should 
have  (see  Diff.  Calc.,  Art.  371) 

du     du  db     du  da      ffi.^       f(  \^a 

277.  Next,  writing  the  integral  in  the  form 

r* 
U=     udx,       (i) 

•  n 


298       .  DEFINITE  INTEGRALS.  [Art.   277. 

as  in  Art.  84,  let  us  suppose  that  the  quantity  u  under  the  inte- 
gral sign  is  a  function  of  some  other  variable  a  beside  the  cur- 
rent variable  x.  Then  U  is  also  a  function  of  a,  we  now  have 

dU 

whence 

d  dU  _du 
da  dx      da' 

Now,  since  differentiation  with  respect  to  independent  vari- 
ables is  commutative  (Diff.  Calc.,  Art.  381),  this  gives 

d   dU     du 

dx  da      da ' 

Hence,  integrating  with  respect  to  x,  we  may  write 
dU      f  du 


da 


in  which  the  constant  of  integration  C  has  a  definite  value  be- 
cause we  have  fixed  the  lower  limit  of  the  integral.  To  find 
this  value  we  notice  that  when  x=a  in  equation  (i),  U=o  inde- 
pendently of  the  value  of  a.  Therefore,  for  this  value  of  x,  U  is 
a  constant  with  respect  to  a,  and  its  derivative  assumes  the  value 
zero.  It  "follows  that,  putting  x  =  a  in  equation  (2),  we  find 
C=o.  Hence  the  equation  may  be  written  in  the  form 


which  expresses  that  an  integral  can  be  differentiated  with  respect 
to  a  quantity  independent  of  the  current  variable  and  the  limits 
by  differentiating  the  expression  under  the  integral  sign. 


§  XIX.]   DIFFERENTIATION   OF  A   DEFINITE   INTEGRAL      299 

If  the  limits  were  also  functions  of  a,  the  total  derivative 
with  respect  to  a  would  also  contain  the  terms  given  in  equa- 
tion (3)  of  the  preceding  article. 

278.  By  means  of  this  theorem,  we  can  derive  from  the  known 
value  of  a  definite  integral  the  values  of  a  series  of  other  integrals. 
For  example,  the  first  of  the  fundamental  integrals,  p.  8,  gives, 
when  n>  —  i, 

1 

(i) 


»  i-  1 


whence,  by  taking  successive  derivatives  with  respect  to  n,  we 
find 

x11  log  x  dx    =  - 


(logx')2dx=- 


and  in  general 


in  which  n  +  1  is  positive,  and  r  is  a  positive  integer. 


Integration  under  the  Integral  Sign. 

279.  Supposing,  as  in  Art.  277,  that  u  is  a  function  of  a  as 
•.veil  as  of  x,  the  definite  integral 

U=    u  dx 


300  DEFINITE  INTEGRALS.  [Art.  279. 

is  a  function  of  a,  and  Uda  may  be  integrated  between  any  limits 
<x0  and  ai  which  are  admissible  in  accordance  with  the  condi- 
tions given  in  Art.  82.  Thus 


£i  r«i  to 

Uda  = 
o  J  aaJ  a 


b 
udx  da. 


We  have  seen  in  §  X.  that  the  order  of  integration  in  this 
double  integral  may  be  changed;  no  change  being  required  in 
the  limits  because  each  pair  is  independent  of  the  other  variable. 
Hence 

r«i  fb  ca 

uda  dx; 


r«i  fb  cai 

Uda  =  | 

J  a0  'a*  aa 


that  is  to  say,  a  definite  integral  can  be  integrated  with  respect 
to  a  new  variable,  between  constant  limits,  by  integrating  the  ex- 
pression under  the  integral  sign  between  these  limits. 

280.  Supposing  the  value  of  U  to  be  known,  we  may  often 
derive,  by  means  of  this  theorem,  the  values  of  other  definite 
integrals.  For  example,  the  integral  employed  in  Art.  278, 

i 
xndx=  --  ,    .......     (i) 

»+i 

is  a  function  of  n;  multiplying  by  dn  and  integrating,  we  have 

dn 


xndn 


fs 
= 

)r 


Performing  the  integrations  with  respect  to  n, 

[*xs-xr  S  +  T. 

~,  -  dx  =  \o&  -  .  .. 
J     lo*  5r  +  i 


§  XIX.]     INTEGRATION    UNDER   THE  INTEGRAL  SIGN          301 

As  a  particular  case,  when  s  =  i  and  r  =  o,  we  have 


x  —  i 

—  dx  =  log2  .......     (4) 

log  X 


Application  to  the  Evaluation  of  Definite  Integrals. 

281.  It  follows  from  the  p  eceding  articles  that,  if  in  the 
case  of  a  definite  integral  to  be  evaluated  we  can,  by  differentia- 
tion or  integration  with  respect  to  a  new  variable,  arrive  at  a 
known  integral,  the  reverse  process  will  give  the  value  of  the 
proposed  integral. 

For  example,  from  Art.  63,  p.  80,  we  derive  the  two  definite 
integrals 

l^  n 

e  mx  sin  nx  ax  =  • 


,  0  m2  +n2 

and 

f00  -***  J  m 

!0e  *   X~m2+n2' 

in  which  m  must  be  positive,  but  n  can  have  either  sign.     Now 
suppose  the  integral  to  be  evaluated  is 


u=\    xe  mxsinnxdx (3) 

J  o 

Here  the  quantity  under  the  integral  sign  is  simplified  by  inte- 
gration with  respect  to  n.     Thus 


udn  =  -      e~mx  cos  nx  dx, 


302  DEFINITE   INTEGRALS.  [Art.  281. 


or,  by  equation  (2), 

f  m 

udn= ^ — 

J  m-  + 


Hence,  taking  derivatives  with  respect  to  n, 


The  "constant  of  integration  "  implied  in  the  indefinite  integral 
of  equation  (4)  is  a  quantity  independent  of  n:  it  need  not  be 
determined,  however,  because  it  would  disappear  in  the  sub- 
sequent differentiation. 

The  proposed  integral,  u  in  equation  (3),  is  in  fact  one  of  a 
series  of  integrals  which  may  be  derived  from  the  known  integrals 
(i)  and  (2)  by  successive  differentiation.  See  example  6  below. 

282.  On  the  other  hand,  the  integral 

-mx  sm  nx 


is  simplified  by  differentiation  with  respect  to  n.     Thus 

dU     (x  m 

—  =      e~mx  cos  nx  dx  =  —,  -  -     ....     (2) 

dn     J0  ,  m2+n2 

by  equation  (2)  of  the  preceding  article.     Hence  by  integration 


mdn  n 

tan-1- 
m 


where  C  is  independent  of  n.  Now  equation  (i)  shows  that  U  =  o 
whenw=o;  hence,  putting  n  =  o  in  this  equation,  we  have 
C=o;  therefore 


-L 


,    .        sin  nx  .  .  n 

U     .  _       

x  m 


§  XIX.]  EVALUATION  BY  DIFFERENTIATION.  303 

in  which  the  tan"1  is  the  primary  value,  so  that  the  integral  has 
the  sign  of  n. 

283.  The  special  case  in  which  m  =  o  deserves  particular 
notice.  Putting  m  =  o  in  equation  (3),  we  find 

(•'"sin  nx'      7t                TZ 
— dx  =  —    or 

Jo        X  2  2 

according  as  n  is  positive  or  negative. 
The  graph  of  the  function 

f  sin  nx 
y= ax. 

*  I      -  'V  ' 

J      Q  *™ 

for  the  case  «  =  i,  is  represented  in  Fig.  3,  p.  no;  the  value  \K 
here  found  is  the  ordinate  of  the  asymptote.  In  the  more  general 
graph  n  is  the  gradient  at  the  origin.  The  asymptote  retains  its 
position  when  n  is  varied,  except  that  when  n  is  negative  it  lies 
below  the  axis  of  x. 

I  will  be  found  that  the  method  of  Art.  282  fails  when  ap- 
plied directly  to  this  definite  integral.  This  results  from  the 
fact  that,  although  n  occurs  in  it,  the  expression  is  not  really 
a  function  of  n. 


Employment  of  Double  Integrals. 

284-.  The  process  illustrated  in  Art.  282  is  equivalent  to 
putting  U  in  the  form  of  a  double  integral  with  both  pairs  of 
limits  constant,  and  then  changing  the  order  of  integration.  For, 
denoting  the  function  under  the  integral  sign  by  u,  so  that 


e  mx  sin  nx 

u=  -        , 

x 


304  DEFINITE  INTEGRALS.  [Art.  284. 

we  found  that 

du 

—  =  e~mx  cos  nx. 

an 

This,  together  with  the  fact  that  «=o  when  n=o,  shows  that  u 
may  be  written  in  the  form 

en 

u  =     e~mx  cos  nx  dn, 

J  o 

and  therefore 

£/=  I    I  e~mx  cos  nx  dn  dx. 


The  order  of  integration  here  indicated  of  course  leads  back 
to  the  original  expression  for  U;  but,  reversing  the  order,  we 
have 


rr 

=          e~mxCQi 

*  o*  o 


U=  |    |    e~mx  cos  nx  dx  dn 

O*   O 

\n  m  dn  m 

' +n2  n° 

285.  The  form  of  a  proposed  integral  containing  no  variable 
except  x  may  suggest  the  introduction  of  another  variable,  in  order 
to  make  this  method' of  evaluation  applicable.  Thus  if  the  value 
of  the  integral 


I  ~j         dx. 
J  o  log  x 


,log 

given  in  equation  (4),  Art.  280,  were  unknown,  its  form  might 
suggest  the  introduction  of  the  variable  exponent  n,  because  differ- 


§  XIX.]          EMPLOYMENT  OF  DOUBLE  INTEGRALS.  305 

entiation  of  xn  as  an  exponential  will  cancel  the  denominator 
log  x.     Thus,  putting 


and  noticing  that  w=o  when  n=o,  we  have 

f1  f';  fM  f1  (n  dn 

U=\   \xndndx=\    \xndxdn=\       —  =log 
JoJo  JoJo  J0n  +  i 


of  which  the  proposed  integral  is  the  special  case  corresponding 
to  n  =  i  . 

286.  Again,  to  evaluate 

f  "log  (i  +cos  x} 

-  dx, 
J0         cos  # 

we  may  introduce  the  variable  z  thus: 

f  "log  (i  +z  cos 

= 

J 


U  =     -  dx. 

COS  X 


The  function  u  under  the  integral  sign  vanishes  when  2  =  0; 
hence,  taking  its  derivative  with  respect  to  z,  we  find  that  U  can  be 
put  in  the  form 

[*(•       dz  fg  f       dx 

U=\       —  •  -  dx=\  -  dz, 

J0]0I+ZCOSX  J0J0I+ZCOSX 


in  which  we  suppose  z<i.     Therefore,  by  formula  (G),  p.  124, 
f*      xdz 


306  DEFINITE  INTEGRALS.  [Art.  286. 

and  putting  z  =  i,  we  have  for  the  particular  case 

flog  (i  +cos  x)          -2 

dx  =  — . 

J  0         cos  x  2 

Transformation  by  Change  of  Variable. 

287.  Examples  of  the  transformation  of  a  definite  integral 
by  a  change  of  the  current  variable  have  already  been  given 
(Arts.  95-98).  The  theorem  of  Art.  97  is  particularly  useful  in 
connection  with  other  transformations.  As  an  illustration,  take 
the  integral 

f* 

M  =      log  sin  6  dd. 

•*  o 
By  Art.  97,  we  have  also 

* 

[~2 
u  =      log  cos  6  dd, 

J  o 

and  adding 

-  - 

f  2       sin  26         i  f*  f  2 

2U=\    log-    —  dO  =  —\  log  sin  <£  d(f>  —  log  2      dd.     .     (i) 

•'o  2  2J0  J0 

But  it  is  easily  shown  that 

JT 
f*  f7 

log  sin  <f)  d<p  =  2      log  sin  d>  d<t>  =  '2U\ 

Jo  Jo 

hence,  substituting  in  equation  (i),  we  derive 

*  — 

f  2  f  2  7T  log  2 

w=      log  sin  ^t//?=      logcos/9^=-       =—1.089.     (2) 

»   r»  *    n 


§  XIX.J     TRANSFORMATION   BY   CHANGE   OF    VARIABLE.     307 

288.  Transformation  of  a  double  integral  may  sometimes  be 
used  to  evaluate  a  definite  integral.  For  example,  let 

r00 

k=\    e~*zdx; (i) 

J  o 

then  we  have  also 

,00 

k  =      e-v*dy. 

*  o 

Since  each  of  the  expressions  under  the  integral  sign  is  inde- 
pendent of  the  variable  contained  in  the  other  and  of  its  limits, 
the  product  of  the  equations  gives 


k2=\    e~*2dx- 


r  rr 

p     y    f]  *\)  =  I  p     (x  -r y  )/7-v  rl^}  t '  f~>\ 

c          u*y —  u./v  u/y    ,      .       \£ ) 

Jo  •  o  •  o 

Regarding  #,  ;y  and  2  as  rectangular  coordinates,  this  double 
integral  represents  the  volume  included  between  the  planes  of 
reference  and  the  surface  whose  equation  is 


Transforming  to  the  polar  coordinates  r  and  6,  where 

x  =  r  cos  6,  y  =  r  sin  6, 

the  same  volume  is  represented  by 

n 

f-     j-30 

£2=  e~r?rdrdd* (3) 

J  o  •'  o 

*  The  analytical  transformation  of  the  double  element  dx  dy  into  r  dr  dO  is 
given  in  Art.  142.  The  integral  in  equation  (3)  represents  the  limiting  value 
when  the  integration  extends  over  a  quadrant  of  a  circle,  and  the  radius  is  then 
made  infinite;  moreover  this  limit  is  found  to  be  finite.  The  integral  in  equa- 
tion (2)  represents  the  limiting  value  when  the  integration  extends  over  a  rectangle 
whose  sides  are  made  infinite.  It  is  clear  that  these  limiting  values  must  be  equal. 


308  DEFINITE  INTEGRALS.  [Art.  288. 

and  integrating,  we  have 


therefore 


289.  A  double  integral  whose  value  is  k2  may  be  formed  in 
a  different  way,  leading  to  another  evaluation  of  k.  Putting 
x=az  in  equation  (i)  of  the  preceding  article,  we  have 

f  00  +  QQ 

k=\    e~a*z*adz=\    e~a*xZadx.  (i) 

v   ' 

•  o  *  o 

Again,  taking  a  as  the  current  variable,  we  may  write 

k=re-°*da 

J  o 

If  the  element  of  this  last  integral  be  multiplied  by  the  con- 
stant k,  the  value  of  the  integral  will  be  multiplied  by  k;  hence, 
using  the  value  of  k  given  in  equation  (i),  we  find 

M.rr  -«*••-•<* 

~LJO€ 

Reversing  the  order  of  integration,  we  have 


--  =  - 

X2+I        4' 


§  XIX.]     TRANSFORMATION   BY   CHANGE  OF    VARIABLE.      309 

hence,  as  before,  k  =  ^7i.     Also,  by  equation  (i), 

k 


f0  k       <Jn 

e-a*X*dx  =  _  =  _V_ 
Jo  «          2d 


See  Art  260. 

290.  The  evaluation  of  an  integral  may  sometimes  be  effected 
by  a  combination  of  the  processes  illustrated  in  preceding  articles. 
For  example,  given 


f« 
=\ 

J  o 


a 

Transforming  by  putting  x=— ,  we  find 

z 

f°°  -(Ti+^dz        f°°  -(*z+^}dx 
u  =  a\    e   u          ~2=a\    e  2'     •     •     •     (2) 

Jo  Z  J o  X 

Now,  differentiating  with  respect  to  a,  we  obtain  from  equa- 
tion (i) 

-^=-20     e   \*    **>  -^; 
aa  J  0  ^2 

hence,  comparing  with  equation  (2), 

dw  du 

-r=—2U        or         — =—2da. 
da  u 

Integrating,  we  have 

logu=-2a+c        or        u=Ae~2a,     ...      (3) 

where  A  is  a  constant  independent  of  a.     To  determine  its  value; 
we  notice  that,  when  a=o,  u  becomes  the  integral  whose  value 


310  DEFINITE  INTEGRALS.  [Art.  290. 

is  found  in  Art.  288;    therefore,  putting  a=o  in  equation  (3),  we 
have  A  =  \^K. 
Hence 

u=\    e  '      **' dx  =  -^— — . 

Jo  2 

Also,  by  equation  (2), 

I    f  •    3  ••  — •• 

.Jo  X  20 

Substitution  of  a  Complex  Value  for  a  Constant. 

291.  If  a  complex  value  is  given  to  one  of  the  constants  in 
an  integral  of  known  value,  the  integral  becomes  a  complex 
quantity,  and  we  may  assume  that  the  real  and  imaginary  parts 
are  represented  by  the  real  and  imaginary  parts  of  the  known 
value.  For  example,  if  in  the  integral 

f  eax 

\eaxdx  = — 
a 


we  put  a=*—m+in  (where  m  is  positive),  and  apply  the  limits 
o  and  oo  ,  we  have 

gtf«  -i*     e~mx(cosnx+isin  nx}"}" 
—  ^-  —r-      — 

0  -m+^n 


}"}" 
, 
_J0 


or 

i          m  +  in 


nx  _j_  i  sn 


m  —  in     m2  +  n2 ' 


Equating  separately  the  real  and  :maginary  parts,  we  have  the 
two  results, 

!"*                             m  f00     m     •                 n 

~m2+n2'  SmHX~m2+n2' 


§  XIX.]        COMPLEX   VALUE   GIVEN   TO  A   CONSTANT.  311 

which   we  have  previously  derived  by  the  method  of  parts,  see 
Art.  281. 

Again,  in  the  equation 

Jo  aa' 

Art.  289,  put  a2=ic2;  whence  a  =  <:(i/|+Vi)>  and  we  find 

f°°/  9     9         •     •          1     2W  I/7T          I  i/7T 

(cos  c2x2 — i  sm  c2x2)dx  =  — :  =  —  — •  (i  —  i). 

J0  C^2   I  +1        2C|/2 

Hence 

,00  /  .00  . 

f  9     9    1  fa  J  f         '          9     9    J  fa 

co'$>c*x2dx= and  — 


.0 

si 

J0 


w«/v  . 

2Cf/2 

and,  if  we  put  y=c2x^, 


Examples  XIX. 

1.  Derive  a  series  of  integrals  by  successive  differentiation  of  the 

r  °°  r°°                11  \ 

definite  integral      e-«*doc.  \    x»e~ax=      —. 

Jo  J  o                          & 

2.  From  the  fundamental  formula  (&')  (p.  9)  derive 

dx  t 


and  thence  derive  a  series  of  integrals  by  differentiation  with  refer- 
ence to  a.  f00       dx  it      1.3...  (2^—3)  _  i 


312  DEFINITE  INTEGRALS.  [Ex.  XIX. 

3.  Derive  a  series  of  integrals  by  differentiating  the  integral  used 
in  Ex.  2  with  reference  to  /?. 

X2n-2dx         -     1.3.5  ...  (2M-3) 


4.  Derive  an  integral  from  that  employed  in  Exs.  2  and  3  by  differ- 
entiating twice  with  respect  to  /?  and  once  with  respect  to  a. 


5.  Derive  an  integral  from  the  result  of  Ex.  II.,  67,  by  differentia- 
tion. f°°  _  dx_ 

Jo  (x2+b2) 

6.  Derive  an  integral  by  a  second  differentiation  with  respect  to  m 
of  equation  (i),  Art.  281;    also,  by  means  of  Ex.  XII.,  18,  Diff.  Calc., 
p.  in,  find  the  result  of  r  differentiations. 

f°°    a    _      _    .  ,  2—     2 

x2e    mx  sin  nx  dx= 
J0 

\ 


r\  f  m~] 

in  nxdx=  --  —  sin     (r+i)cot~J—     . 


7.  Derive  an  integral  by  differentiating  equation   (i),   Art.    281, 
with  respect  to  n.                                     f"3  m2—n2 

I         *v*>  rVIX   r+r\c*     11  -\-  -  _  _ 

Jc 


OCC     '        COS  HOC  —  • 


^4   second  differentiation  gives  again  the  result    of  Ex.   6. 
8.  From  the  definite  integral 

»rf\ 

m 


e~mx  cosnxdx= 


m2+n2 


derive  the  result  of  Ex.  7  by  differentiation  with  respect  to  m,  also 
an  integral  by  a  second  differentiation. 

2m(m2—  3«2) 


f  o 

^ 

J 


-  j 

»«*  cos  nx  ax= 


§  XIX.]  EXAMPLES. 


The  corresponding  general  integral,  which  is 

xre~mx  cos  nx  dx= : — -r-  cos     (r+ 1)  cot"1— 

J0  —  L  n  J 

(w2  +  n2)   • 

be  derived  directly  from  Ex.  6  by  differentiation  with  respect  to  n. 

9.  Derive    an    integral  by  integrating          2       2= — . 

"J  o  a  +  #       20 

f00 F  P  q~]dx      K  .       p 

tan"1- — tan"1—     —  =  — log—. 
J0  L  x  x  _]  x       2      °  q 

10.  Derive  a  definite  integral  by  integrating 

I"00  n 

a  —  m%  cin  -M-V  //'y=  

3111    «•.*    U.^-  „  ™ 

J0  m^  +  n* 

with  respect  to  w. 

(cos  ax—  cos  fct)d#= —  log  —75 s. 

1T  *>  WJL*— I— /T* 

Jo-*  •*  '"      I    <* 

11.  Derive  an  integral  by  integrating 

r  m 

a — 1tlX   f*r\G.   WV  /7^v 

1_>J3    «^/   U..V  „  „ 

Jo  m^+n2 

with  respect  to  w. 

r°°g-a* e  —  bx  j  J2^.W2 

cos  nx  dx=  —  log  — 5. 

J0          ^  2      &  a^  +  n2 

Each  of  the  integrals,  for  which  Exs.  10  and  n  gwe  the  differences, 
of  values  corresponding  to  different  constants, is  separately  infinite.  The 
two  results  together  give  the  more  general  difference  formula 

e cos  ax    e ^LJ^=  $  jog 


j0  -  m2  +  a2' 

12.  Derive  an  integral  by  integration  from  the  result  of  Ex.  II.,  67. 

p(g+b) 

q(p+bY 


f°°i  r  *  a~]     dx         n 

tan-'^-tan"1^-      2  ,  L2=-l2 
J0rx;L  x  xjx2  +  b2     2b2 


314  DEFINITE  INTEGRALS.  [Ex.  XIX. 

13.  Evaluate  the  integral      4  log  (i  +  tan  (f)),  using  the  theorem  of 
Art.  97,  J  °  n  log  2 

f  ""^  log  x  dx 

14.  — 9^ — 9-9-. 

J  O     \^     "1"  ^     J 


8 
log  a 


tan     x  *  ax  TT 

I5'J, 


f 
. 
J 


16.        tan-1  — 


.  _ 

a  ^4+a4  i6a2 


«»oo  r~  2  "1 

18.        log     i  +  ^  hog  x  dx.  Tra(loga-i). 

Jo          L.        x  ~J 

19.  f1^ 

Jo          I 
20. 


21.   Prove  that 


xa~Icos  (b  log  #)d#=  -27rr2     and         ^  ~  x  sin  (6  log 
Jo  d  -rtr"  J0 

f0 

u= 

J 


22.  Evaluate  u=      e~a*x*  cos  2rxdx. 


du  l  ,/r  _ 

Integrate  -j-  o^  ^arfo.  u=—  —  e 

1 


T,  Ai-  f°°a  cos  bx    , 

23.  Futtmg  M=       g2       2    a#,  prove  by  a  double  integration  by 


§  XIX.]  EXAMPLES.  315 

d2u 

parts  that  b2u=—  -5,  which  is  satisfied  by  u=Aeab+Be~ab.     Thence 
da^ 

show  that,  when  a  and  b  are  positive, 

cos  bx  ,        TZ        , 
-dx=—e~ab. 


24.  Derive  integrals  by  differentiation  and  integration  of  the  result 
of  Ex.  23;  and  thence  deduce  the  equation  of  Art.  283. 

p*  sin  ft*          *  f    sinftag    dx=^ll_e-ab] 

J0  a2+*?d-       26  )oX(a2+X2r*     2a*[I 

.  r°°cos  bx  dx  - 

25.  Evaluate  the  integral        ,2+x2\2-  ~se~ 


XX. 

Infinite  Values  of  the  Function  under  the  Integral  Sign. 

292.  We  have  seen  in  Art.  82  that  when  f(x]  is  a  real  and 
finite  one-valued  function  for  all  values  of  x  between  and  including 
the  values  a  and  b,  the  integral 

f* 

f(x)  dx 

J  a 

has  a  real,  finite  and  definite  value.     In  fact,  under  these  circum- 
stances, the  graph  of  the  indefinite  integral,  that  is  the  curve 


y= 


31  6  DEFINITE  INTEGRALS.  [Art.  292. 

(see  Art.  85  et  seq.}  is,  for  this  range  of  values  of  x,  determined 
by  the  fact  that  it  passes  through  the  point  (a,  o),  and  that  its 
gradient  is  given  at  every  point  by  the  equation 


-==. 

Now  when  the  value  of  f(x]  is  infinite  at  the  limit  the  integral 
itself  may  increase  without  limit:  but  not  necessarily  so,  as  illus- 
trated by  the  graphs  in  Figs.  5  and  6,  pp.  112  and  113,  where 
in  each  case  the  kn  wn  value  of  the  indefinite  integral  shows 
that  it  is  finite  at  the  critical  points  where  f(x]  is  infinite.  Thus 
we  can,  for  example,  write 

f1       dx  x 


although  f(x)  is  infinite  at  the  upper  limit. 

293.  When  the  definite  integral  is  regarded  as  the  limit  of  a 
sum  (see  Art.  99)  an  integral  of  this  kind  is  generally  characterized 
by  the  fact  that  the  extreme  elements  of  the  sum  vanish  when 
we  pass  to 'the  limit  and  the  number  of  elements  becomes  infinite. 
Consider,  for  example,  the  integral 

n 

\    log  tan  (j>  d(f>, 

J  o 

in  which  log  tan  0  is  infinite  at  the  lower  limit.  Here  the  first 
element  of  the  sum  of  which  the  integral  is  the  limit  is  4(f>  log  tan  J<£. 
Passing  to  the  limit  when  J<£  vanishes,  and  writing  z  for  J<£,  this 
element  takrs  the  indete  minate  form  z  log  tan  z]z=0,  the  value 
of  which  is  found  on  evaluation  to  be  zero.  Thus  the  given  in- 
tegral is  the  sum  of  an  infinite  number  of  vanishing  elements 
and  admits  of  a  finite  value  just  as  in  the  ordinary  case.  Com- 
pare the  integral  evaluated  in  Art.  287. 


§  XX]     CAUCHY'S  GENERAL   AND  PRINCIPAL   VALUES.         317 


Cauchy  s  General  and  Principal  Values, 

294.  In  general,  if  f(x)  is  infinite  only  at  the  upper  limit  b, 
the  integral  may  be  regarded  as  the  limiting  value  when  e  is 
diminished  without  limit  of 

rb  —  e 

/(*)  dx 
•  a 

(where,  supposing  b>a,  e  is  a  small  positive  quantity),  and  this 
may  have  a  finite  or  an  infinite  value. 

In  like  manner,  when  f(x)  is  finite  for  a  and  b  and  for  all 
intermediate  values  except  the  single  value  c,  for  which  it  is 
infinite,  Cauchy  regarded  the  integral  as  the  limit  of 


fc-iie  rb 

/(#)  dx  +        f(x)dx, 

J  a  •'  c+ve 


when  e  decreases  without  limit,  //  and  v  being  positive  numbers. 
If  both  parts  of  this  expression  have  finite  limits,  the  integral 
h;;s  a  finite  value.  If  both  parts  become  infinite  with  the  same 
algebraic  sign,  the  integral  is  infinite;  but,  if  they  become 
infinite  with  opposite  signs,  the  re  ult  takes  the  indeterminate 
form  oo  —  oo  ,  and  may  have  a  finite  value. 

In  this  last  ca  e,  the  limiting  value  r,  generally  found  to 
depend  upon  the  ratio  fjt  :  v.  This  was  called  by  Cauchy  the 
general  value  of  the  integral;  while  the  special  value  assumed 
when  n  =  v  he  called  the  principal  value  of  the  integral. 

295.  For  example,  if  a  and  b  stand  for  positive  quantities, 


[b    dx 
f(x)  in  the  integral         —  is  infinite  for  the  single  intermediate 

J  —  a  X 

re 

[""'dx     fb  dx  ue  b 

—  +      —  =log—  +log— » 
J  _„  x      }vfx         &  a        &  ve 


va  ue  x=o.     Here 


318  DEFINITE   INTEGRALS.  [Art.  295, 

which  takes  the  form   —  <x>  +00    when  e=o;    but,  by  algebraic 
reduction,  the  expression  becomes 

b  a 

log -  + log-, 

which  is  accordingly  the  general  value  of  the  integral,  and  putting 
jj.  =  v,  we  have  for  the  principal  value  log  b  —  log  a. 


Integrals  with  Infinite  Limits. 

296.  An  integral  with  an  infinite  upper  limit  is  the  limit  of 
an  integral  of  the  form 


AI  — —  I     /f'vA  /7'v*  /T \ 

7          -/W  ax> W 

Ja 

wrhen  x  is  increased  without  limit.  In  order  that  it  may  have  a 
finite  and  definite  value,  the  graph  of  the  indefinite  integral, 
represented  by  equation  (i),  must  have  an  asymptote  parallel  to 
the  axis  of  x.  When  this  is  the  case,  it  is  necessary  that,  if  f(x) 
(which  is  the  gradient  or  value  of  tan  0  in  the  curve)  approaches 
to  a  definite  *  limit,  that  limit  shall  be  zero. 

297.  On  the  other  hand,  /(#)  may  approach  zero  as  a  limit 
when  #=oo,  and  yet  the  integral  may  increase  without  limit. 
For  example,  in  the  integral 

'dx 

y=' 

*  If  (x)  does  not  approach  a  definite  limit,  the  integral  may  remain  finite  as 
x  increases  without  limit  and  yet  not  have  a  definite  limiting  value.     For  example. 

I* 

sin  x  has  no  definite  value  when  x  =  °o,  and       sin  x  dx  approaches  no  definite 

J  o 
limit  as  x  increases,  but  its  value  must  lie  between  o  and  2. 


§  XX.]      INTEGRALS  WITH  INFINITE  LIMITS,        319 

f(x}=o  when  x=<x>,  but  the  integral  (whose  value  is  log  x)  be- 
comes infinite.  The  infinite  branch  of  the  graph  in  this  case 
tends  to  parallelism  to  the  axis  of  x,  but  the  branch  is  parabolic, 
and  y  increases  without  1  mit. 

When  the  integral  is  regarded  as  a  sum,  the  firs  case  is  analo- 
gous to  an  infinite  series  of  terms  decreasing  without  limit  and 
having  a  finite  sum,  and  this  last  case  is  analogou  to  a  series 
(like  that  given  in  Art.  180,  Diff.  Calc.,  p.  176)  in  which  the  terms 
decrease  without  limit,  but  the  sum  nevertheless  has  no  limit. 

298.  When  f(x]  is  real  and  finite  for  all  real  values  of  x, 
and  approaches  zero  both  for  x=<x>  and  x=  —  oo,  the  graph  of 
the  integral  may  approach  an  asymptote  at  each  end.  In  this 
case  the  integral 

C 

/(#)  dx, 


has  a  finite  value,  as  illustrated  by  Fig.  7,  p.  115. 

Again,  when  the  indefinite  integral  becomes  infinite  both  for 
positive  and  negative  values,  so  that  both  parts  of  the  definite 
integral  become  infinite,  the  integral  may  have  a  finite  value,  if 
these  parts  become  infinite  with  opposite  signs.  In  this  case,  the 
integral  must  be  regarded  as  the  limit  of 


(iA 

J  ~ 


f(x)  dx, 


where  /*  and  v  are  positive  numbers  and  h  increases  without 
limit.  This  limit,  like  that  considered  in  Art.  294,  is  in  general 
dependent  upon  the  ratio  //  :  v,  and,  by  assuming  {i=v,  we  can 
obtain  a  "principal  value."* 

*  In  this  case,  if  the  negative  branch  of  the  graph  were  folded  over  on  the 
axis  of  y,  the  principal  value  would  be  the  limit  of  the  vertical  distance  between 
the  two  branches. 


320  DEFINITE  INTEGRALS.  [Art.  299. 

Integrals  of  Certain  Rational  Fractions. 

299.  Let  us  take,  for  example,  the  integral  corresponding  to 
a  pair  of  imaginary  roots  cc±//?  in  the  decomposition  of  a  rational 
fraction  (Art.  18).  It  will  be  convenient  to  put  the  quadratic 
fraction  in  the  form 

(i) 


(x-a)2+p2 
because  this  is  the  sum  of  the  linear  partial  fractions 

A+iB         A-iB 

x—a+ifl     x—a—ijf 

Thus  we  have  to  consider  the  integral 


dx. 

TC  — ay  -tp" 

The  indefinite  integral  is 

A  log[(*-a)2+/?2]+25tan-'^^ 

The  second  term  is  finite  at  each  limit,  and  the  value  when 
the  limits  are  applied  is  2B-rt  But  the  first  term  is  infinite  at 
each  limit,  hence  its  value  is  the  limit,  when  h  is  infinite,  of 


*     2A(x  -a) 

~A  log  7-:—  —  ^—50          =2A  log  —  . 
&  22  3 


.        N2  ,  ,3, 
J  _,;,(*-  a)2  +/22 

Thus   h    general  value  of  the  integral  is 


and  the  principal  value  is  2  ETC. 


§  XX.]  INTEGRALS  OF  CERTAIN   RATIONAL  FRACTIONS.     321 
300.  Let  us  suppose  that,  in  the  integral 

*) 
-?dx> 


I 


<j>(x)  is  a  rational  integral  function  of  which  x2n  is  the  highest 
term,  while  f(x)  is  a  polynomial  of  lower  degree;  also  that  <j>(x)  =o 
has  no  r,  al  roots.  The  fraction  can  be  decomposed  into  n  quad- 
ratic fractions  corresponding  to  the  n  pairs  of  imaginary  roots. 
Thus 


f(x)  ^2Al(x-al')+2Bipl  2An(x-an)  +2Bni8n 

<j>(x)          (x-atf+p?  (x-ajt+pj 

Hence  by  equation  (3)  above  we  have 


Thus,  in  general,  the  integral  is  infinite  at  both  limits  and  is 
indeterminate  in  value. 

Let  us  now  further  suppose  that  /"(:*;)  is  al  least  two  units  lower 
in  degree  than  <f>(x).  Then,  equating  the  coefficients  of  x2n~l  in 
the  result  of  clearing  equation  (i)  of  fractions,  we  have  c  =  2lA. 
Therefore,  in  this  case,  we  have 


the  indefinite  integral  being  now  finite  at  each  limit. 
301.  Let  us  apply  this  result  to  evaluate 


where  m<n.    The  roots  of  the  equation  x2n  +  i  =o  are  the  values 


322  DEFINITE  INTEGRALS.  [Art.  301. 

i 
of  (cos  7i  +i  sin  TT)^M,  which,  by  De  Moivre's  Theorem,  are  of  the 

form 

(2^  +  1)71  (2^  +  1)7: 

sin 


2H  2H 


where  k  has  the  2n  values,  o,  i,  .  .  .  .  ,  2n  —  i.     These  are  equiva- 
lent to  the  n  conjugate  pairs, 


x  =  cos  -  ±  «  sin 


2H  2H 

where  k  has  the  n  values  o,  i,  .  .  .  ,  n  —  i.  Now,  if  he  values  of 
A  and  B  corresponding  to  a  pair  of  roo  s  are  denned  by  expres- 
sion (2),  Art.  299,  A—iB  will  be  he  numera  or  of  the  partial 
fraction  corresponding  to  a+ifi.  By  equation  (3),  Art.  21,  the 
value  of  this  numerator  is 


Hence,  in  the  present  case, 


because  (a  +i/3)  2n  =  —  i  .     Hence 

i  f       2&  +  i  2k  +  i  H2»«+i 

Ak  —  iBk=  --    cos  --  7r+«sm  -  TT  , 

2U\_  2H  2H        J 

or,  using  DeMoivre's  Theorem,  and  putting  for  abridgment 

2W4-I 


2tl 


Ak-iBk= [cos  (2k  +  i}6+i  sin 


§  XX.J  INTEGRALS  OF  CERTAIN   RATIONAL  FRACTIONS.     323 

Thus  zBk  =  -  sin  (2&  +  i)0,  and  by  equation  (3),  Art.  300, 
ctoc ==-  ~~\  sin 

OC2        I     T  M, 


sn  3+  '  •  '  +  sin  (2W-I)#]- 
This  expression  may  be  put  in  a  more  compact  form;  for,  if 

5"  =  sin  (9+  sin  3$+  .  .  .  +  sin  (2^  —  1  }6, 
we  have 

25  sin  0  =  2  sin2  0+2  sin  0sin  30  +  ...  +2  sin  6  sin  (2^  —  1)0 

=  i-cos  20+cos  2/9  -cos  4/9+  ...  +cos  (2n-  2)6  -cos  2nd 
=  i—  cos  2W  =2  sin2  «(9  =  2  sin2  (w+|)7:  =  2. 

Therefore  5  =  cosec  0,  and 

f*      ^2W  ic  2m  +  1 

-—  —  d#=—  cosec  -  TT  .....     (i) 

J-oo^  +  i         n  2n 

302.  Since  the  function  under  the  integral  sign  is  unchanged 
when  we  change  the  sign  of  x,  it  obviously  follows  from  this 
result  that 

f°°     X™  71  2M  +1 

—  —  —  dx  =  —  cosec  -  T:  .....     (i) 

J0^2n  +  I  2W  2W 


This  equation  admits  of  some  noteworthy  transformations.    In 

_i_  i  _ 

th    first  place,  putting  y  =  x™,  whence  x=y2n,  dx  =  —  y™   *  dy, 

2H 


we  have 


i  f  v  2n  it 

—     -  ay  =  — 

2MJ-        I  +J  2U 


2m  +i 
cosec  -  it\ 

2H 


324  DEFINITE  INTEGRALS.  [Art.  302. 


2W  +  I 

or,  putting  =r, 


i+y      sin 


which  contains  a  single  constant  r  in  place  of  m  and  w.  Since 
w  and  n  are  positive  integers,  and  m<n,  the  value  of  r  must  lie 
between  o  and  i;  but,  by  the  law  of  continuity,  it  is  otherwise 
unrestricted,  because  it  can  be  made  to  approach  a  given  value 
(commensurable  or  incommensurable)  as  near  as  we  choose,  by 
taking  sufficiently  large  values  of  m  and  n. 

Since  sin  (r  —  i  )?r  =  sin  nr,  we  may,  by  putting  r  in  place  of 
j—  r,  put  equation  (2)  in  the  form 


(3) 


+y)yr 


In  this  equat'on  also  we  must  have  o<r<i. 

Again,  putting  y=xp  in  equation  (2),  p  being  positive,  we  have 


\~ 

or,  putting  pr—i=q, 

f°    &  TT  q  +  i 

-dx  =  —  cosec  — — TT, (4) 


J0i+*>         p  p 

in  which  p  is  positive,  and  q  must  lie  between  —  i  and  p  —  i. 
This  equation  in  fact  includes  equation  (i),  which  is  therefore 
not  restricted  to  integral  values  of  m  and  n. 

As  a  particular  case,  we  may  put  q=o,  provided  p>i;   thus, 

when  p>i, 

f°°    dx       TT  it 

>=7cosec^ (s) 


§  XX.]  FRULLANI'S  INTEGRAL.  325 

Frullani's  Integral. 
303.  Suppose  that  an  integral  can  be  put  in  the  form 


'•dx, 


x 


in  which  c  is  positive,  and  that  while  'x  varies  from  c  to  infinity 
<f>(x)  does  not  become  infinite,  but  retains  he  same  sign  (say  the 
positive)  and  approaches  a  finite  limit  not  zero;  then,  although 
the  function  under  the  integral  sign  approaches  zero,  the  value 
of  the  integral  will  be  infinite  when  x  =  oo .  For,  let  A  be  the 
least  value  of  $(x]  for  the  entire  range  of  values  of  x,  then 


because  every  element  of  the  given  integral  is  not  less  than  the 
corresponding  element  of  the  integral  in  the  seccnd  member. 
But,  when  x  increases  without  limit,  the  value  of  the  integral  in 
the  second  member,  whi.h  is  A  (log  x  — log  c),  becomes  infinite; 
hence,  a  fortiori,  the  value  of  the  given  integral  is  infinite. 

304.  But  the  difference  between  two  integrals  of  the  form 
considered  may  be  finite.  Consider,  for  example,  Frullani's 
Integral, 

r$(ax}-$(bx) 
U  =  —  ax, 

J  o  ^ 

in  which  a  and  b  are  positive,  and  <j>(x]  does  not  become  infinite 
for  any  positive  value  of  x.  We  notice,  in  the  first  place,  that, 
supposing  0'(o)  not  to  be  infinite,  zero  is  admissible  as  a  lower 
limit,  because  when  x  =  o  the  quantity  under  the  in  egral  sign  is 
found  on  evaluation  to  have  the  finite  value  (a— 6)$'(o). 


326  DEFINITE  INTEGRALS.  [Art.  304. 

Employing  the  method  of  Art    284,  we  have 

f  °°  f°  fa  f°° 

U=\        (j>f  (ax)da  dx  =  \        <j>'  (ax}dx  da. 

JoJb  Jb  Jo 


Now 


a 
or 


therefore 

f*V(<Ec)-0(fo)  a 

dx  =  \<f)(cQ  )  —  <i(o)l  log  -r.  (i) 

/v  L/V/rV/JOA  \/ 

Jo  * 

Fo:  example,  if  ^>(^)=tan~r  x,  0(oo  )  =  $n  and  ^>(o)=o,  therefore 

-dx  =  ^-\og^. 


0  x  2 

Compare  Ex.  XIX.,  9. 

305.  When  <f>(x)  is  a  function  which,  although  remaining 
finite,  has  no  definite  limiting  value  when  x  =00,  equation  (i) 
fails  to  determine  the  value  of  he  in  egral.  Fo  example,  when 
<£(#)=  cos  #,  cos  oo  has  no  definite  value.  The  following  mode 
of  investigating  the  in  egral  will,  however,  show  that  in  these 
cases  equaton  (i)  will  hold  true  if  for  <£(«>)  we  substitute  the 
mean  value  of  <{>(x)  over  an  infinite  ange  of  values  of  x. 

For  this  purpose,  we  put 


-i: 


•dz, 


an  integral  having  a  finite  value  when  h  is  finite.      Putting  z  =  ax, 
we  find 

j>(ax)-^(o)j 

dx (i) 


§XX]  FRULLANl'S  INTEGRAL.  327 


In  like  manner, 

h_ 

_  rb<f>(bx)-<f>(o) 
u  —  I 

Jo 


dx. 


X 

Supposing  a>b,  this  last  equation  can  be  written  in  the  form 


h 


_ 

fad>(bx)—d>(o)  f 

vi    )    vv  Jdx  i 

J0                X  J 


h       X 


Equating  the  values  of  u  in  equations  (i)  and  (2),  we  have 


Frullani's  Integral  is  the  limit  when  h=cc  of  the  integral  in 
the  first  member.  As  to  that  in  the  second  member,  let  us  first 
suppose  that  the  mean  value  of  <f>(x)  over  the  infinite  range  of 

values  of  x  is  zero.     Since  the  greatest  value  of  the  factor  — 

a 
under  the  integral  sign  is  j-  (which  is  its  value  at  the  lower  limit), 

a 


h       h 

Now  the  mean  value  of  <j>(x)  over  the  range  of  values  —  to  r  is 

a      b 


b      a   a 
hence,  if  the  limiting  value  of  M  is  zero,  that  of  the  integral  in 


328  DEFINITE  INTEGRALS.  [Art.  305. 

the  second  member  of  the  inequality  (4)  is  zero;  and,  a  fortiori, 
that  of  the  first  member  is  zero.  Substituting  in  equation  (3), 
and  then  making  h  infinite,  the  result  is  the  same  as  if  0(°o)  in 
equation  (i),  Art.  304,  were  equal  to  zero.  For  example,  putting 
<£(#)=  cos  x,  the  limiting  value  of  M  is  zero,  and  we  have 

f°°cos  ax  —  cos  bx  b 

-  ax  =  log  —  . 
J0  x  6  a 

Finally,  if  the  mean  value  of  <j>(x)  for  an  infinite  range  of 
values  of  x  is  the  finite  quartity  M,  we  may  put 


in  which  </>(x)  is  a  function  having  zero  for  its  mean  value.     Sub- 
stituting in  equation  (3),  and  then  making  h  infinite,  we  derive 


-,    ...      (4) 


where  M  takes  the  place  of  <K°°)  in  equation  (i),  Art.  304. 

306.  It  may  be  remarked  that  when  </>(oo)=o,  and  also 
when  its  mean  value  is  zero,  the  two  parts  of  Frullani's  Integral 
are  not  each  infinite  at  the  upper  limit  (as  they  were  under  the 
conditions  named  in  Art.  303).  In  like  manner,  if  0(o)=o, 
they  are  not  infinite  at  the  lower  limit.  Thus,  if  both  these 
conditions  hold,  both  zero  and  infinity  are  admissible  limits 

f  (j)(cix)dx 
of  the  single  integral     —  -  .     But,  in  this  case,   he  second 

j        x 

member  of  equation  (4)  vanishes,  and  we  have 


I  dx  —  I 

Jo        X  J0         X 


§  XX]  FRULLANI'S  INTEGRAL.  329 

It  follows  that  an  integral  of  this  form  has  a  value  independent 
of  a,  so  long  as  a  i,  positive,  which  has  been  assumed  in  our 
demonstration.  This  might  also  have  been  inferred  from  the 
result  of  substituting  z  for  ax.  But  even  in  this  last  process  we 
cannot  infer  that  the  value  is  the  same  for  positive  and  negative 
values  of  a,  because  the  integral  ceases  to  have  a  meaning  when 
a  by  gradual  change  of  value  passes  through  the  value  zero. 
An  integral  of  this  form  may  therefore  be  a  discontinuous  func- 
tion of  a.  For  example,  we  have  seen  in  Art.  283  that 

f°°sin  ax          n  K 

-  dx  =  —    or     — 

Jo        x  2  2 

According  as  a  is  positive  or  negat  ve. 

Integrals  obtained  by  Expansion. 

307.  We  have  seen  in  Art.  78  that,  when  .he  integrat'on  of 
j,c(  )dx  cannot  be  effected  in  finite  terms,  the  integration,  term 
by  term,  of  its  development  in  powers  of  x  will  give  the  like  de- 
velopment of  he  integral.  Thus,  if 


we  have,  by  development  of  the  logarithm, 

f  /       x    x*          \ 
0(*)  =  Jo^i  +-+j+...  )dx; 

whence,  integrating  term  by  term, 

2      X3      3(4 

+2+-2  +  ........     (2) 


330  DEFINITE  INTEGRALS.  [Art.  307. 

This  defines  (j>(x)  by  a  series  which  is  convergent  when  x<  i. 
It  ie  also  convergent  when  x=i,  its  value  (see  DifT.  Calc.,  Art.  238) 
then  being  \x2.  Thus 

,,  .      [\         i     dx    r? 
d>(l)=\   log-          —  =—  ......      (?) 

v  '     J0    6  i  -x  x      6 

308.  In  like  manner,  the  development  of  the  quantit  under 
the  integral  sign  in  any  other  form  of  infinite  series  may  admit 
of  integration  term  by  term.  For  example,  if  we  transform  the 
integral  considered  in  the  preceding  article  by  putting  x  =  i—z, 
we  have 


* 

and,  expanding  the  factor  (i-z)"1  in  a  series  convergent  when 
z  is  between  o  and  i, 

<f>(i  -z)  =  I  log  z(i  +z  +z2  +.  .  .  )(fe.      ...     (4) 

J  i 

Each  term  under  the  integral  sign  is  of  the  form  zn  log  z  dz,  and 
integrating  by  parts  we  have 


zMlog 
J  i 


zM+Ilogz         zw+1  i 

zdz=  — 

n  +  i 


Giving  to  n  the  values  o,  i,  2,  etc.,  and  substituting  in  equa- 
tion (4), 

z2     z3  n      r      z2     z3 

-  +-+- 


§  XX.]  INTEGRALS  OBTAINED   BY  EXPANSION.  331 

The   coefficient  of  Iog2  is  the  series  for   —log  (i—  z),  hence 
the  equation  may  be  written 


<£(i  -  z)  =  -  log  z  log  (i  -  z)  -  c/>(z)  +  <£(i  ). 
Putting  x  for  z,  we  have  the  relation 


...     (5) 


as  a  property  of  this  function.     As  a  particular  result,  putting 
x  =  %,  we  find 

ft  I       dx_X*       (lQg2)2 

°  I  —X   X        12  2 

309.  In  the  case  of  a  definite  integral,  the  result  of  expansion 
in  a  series  of  integrable  form  is  in  general  a  numerical  series, 
by  means  of  which,  if  convergent,  the  value  may  be  computed; 
while  in  some  cases  the  sum  may  be  already  known.  For  ex- 
ample, to  find 

r  n  oc  sin  oc  doc 

U=L~< 


where  p  is  a  positive  integer.      Expanding    (i+cos2^)"1,   we 

have 

ft 
U=\  xsin  x(i  —cos2t>x+cos4px  —  cos6px+. . .  }dx. 

J  o 

Integrating  the  typical  term  by  parts, 

r«  #cos2rH"I:xn'r         T       f" 

x  cos^x  smxdx= +  — r— -    cossrf+Ix  dx 

J0  2rp  +  i     J0    2  p  +  iJ0 

It 
~2rp+i* 


332                                     DEFINITE  INTEGRALS  [Art.  309. 

since  the  final  integral  vanishes.     Hence,  giving  to  r  the  values 
o,  i,  2,  etc.,  we  hav 

f*xs'mxdx       /iii  \ 

J0i+cos2^~    \       2^  +  1^4^  +  1     6/>  +  i  / 


When  p=i,  the  series  is  the  result  of  putting  x  =  i  in  Gregory's 
series  for  tan"1  x;   thus,  as  a  particular  case, 

("X  sin  x  dx        /       i      i      i  \     n2 

=  -   i  —  +—-  +  ...      =  -. 


Joi+cos2*       \      3     5     7         '/     4 

Series  in  Sines  and  Cosines  of  Multiple  Angles. 

310.  When  A  is  numerically  greater  than  B,  the  expression 
A  +B  cos  #  can  be  reduced  to  the  form  m(i  +20.  cos  x+a2),  where 
a  is  real  and  may  be  taken  less  than  unity.  (See  Art.  313.) 
Now  the  expression  i  +20  cos  x  +a2  admits  of  conjugate  imaginary 
factors,  thus 

i  +20,  cos  x+a2  =  (i  +a  cos  x+aistn.  x)(i  +a  cos  x  —  ai  sin  x), 
or,  using  the  exponential  notation  (Diff.  Calc.,  Art.  222), 

cos  x+a2  =  (i  +aeix)(i 


The  sums  and  differences  of  the  expansions  of  like  functions 
of  these  imaginary  factors  may,  by  means  of  the  equations 

2  cos  x  =  eix+  e~ix    and     21  sin  x  =  eix  —  e~ix, 

be  expressed  in  terms  of  the  sines  and  cosines  of  the  multiples  of 
the  angle  x,  thus  giving  rise  to  a  variety  of  developments  in  mul- 
tiple angles.  Among  the  simplest  are  the  series  deduced  below. 


§  XX.]  MULTIPLE-  ANGLE  SERIES.  333 

311.  From  the  expansions 

=  i  -  aeix  +a2e2ix  - 


i  +ae   *x 

we  have  by  addition 
2  +20,  cosx 


121 


=  2|i  -a  cos  x+a2  cos  2X  -a3  cos  T.X+.  .  .  .1; 


i  +2a  cos 

whence,  subtracting  unity  to  simplify  the  numerator, 
i  -a 


-2 

=  !  —  2d  COS#  +  2#2COS  2X  —  2  a3  COS  1.X  +  .  .  . 


Again,  the  difference  of  equations   (i)   and    (2)   gives,  after 
dividing  by  2ai, 

sin  x 

=  sm  x—  a  sin  2X+a2  sm  T.X  —  a3  sin  4^+.  .  .      (4) 


i  +20,  cos  x+a* 

These  series  are  convergent  when  a<i. 

In  like  manner,  from  the  logarithmic  series, 

a2          a3    . 

log  (i  +ae*x)=ae?x e2tx+—e31-x  — .  .  . . 

2  3 

a2  a3 

log  (i  -\-ae~  w)=ae~tx e~2tx  -{ — e~3*x4 

2  3 

we  have,  by  addition, 

|~  a2  a3 

-2^a  cos  x-  — 


"I 
— ...    .    (5) 


334  DEFINITE  INTEGRALS.  [Art.  311. 

Again,  by  subtraction,  we  derive  from  the  same  equations 

i  +00*  a2   .  a3  ~\ 

log 7-  =  21  \  a  sin  x sin  2#H —  sin   T.X  —  ...    ,     (6) 

&i+ae~tx        L  2  3  J 

To  reduce  the  first  member  to  the  pure  imaginary  form,  we  have 

i  +aetx      i  +a  cos  x  +  ai  sin  x 
i  +ae~tx     i  +a  cos  x  —  ai  sin  x ' 

in  which  let  us  put  i  +a  cos  x=p  cos  y,  a  sin  x=p  sin  y,  so  that 

a  sin  x 

tan  y  = . 

i  +a  cos  # 

We  shall  then  have 

i  +aetx  cos  y+i  sin  -y  e** 

logn^^=1°g 


cosv-isn  y 
Hence  equation  (6)  becomes 

a  sin  x  a2   ,  a3   . 

/y  =  tan~I  -  =  a  sin  ^  —  -  sin  2^+—  sin  ^  —  .  .  .       (7) 
i  +a  cos  x  23 

A  discussion  of  this  equation  will  be  found  in  Art.  324  et  seq. 

Integrals  Developed  in  Multiple  -angle  Series. 

312.  The  direct  integration  of  equation  (3)  of  the  preceding 
article  between  o  and  x  gives,  when  a<i, 


_  2Ja 

a  JJ, 


dx 


i  +2<zcos  A;+a2 

f  «2    .  «3   .  "1 

=#  —  2    a  sin  x sin  2X+—  sin  ^#— . . .        .     .     (i) 

23  J 

in  which  the  series  is  convergent. 


§  XX.]          INTEGRALS  IN   MULTIPLE-ANGLE  SERIES.  335 

An  expression,  like  that  in  the  second  member,  which  is  periodic 
except  for  one  term  of  the  first  degree  in  the  independent  variable, 
is  sometimes  called  a  quasi- periodi  function.  The  integral  of 
a  periodic  functi  n  goes  through  the  same  states  of  increase 
or  decrease  in  corresponding  parts  of  successive  periods  of  the 
independent  variable.  Hence,  except  when  the  increment  corre- 
sponding to  a  complete  period  vanishes,  such  an  integral  must 
have  this  quasi-periodic  character.* 

In  the  presen  case,  the  series  vanishes  for  o  and  2n,  and  also 
for  the  half-period  TT,  so  that  we  have 


i: 


dx  7t 

....   (2) 


i+2acosjc+a2     i  —  a2' 


When  x  =  \n,  the  terms  containing  even  powers  of  a  in  equa- 
tion (i)  vanish,  and  we  have 

n 

dx  x        ~      a      a5    a7 


whence,  by  Gregory's  series, 

a 

[~  dx  i  1 

—  —  2  =  -  i   --  2  tan-1  a   .     .     .     (3) 
,  J0  i  +20.  cos  x+a2     i  —  a2l_2  J 

313.  When  a>i,  in  the  integral  of  equation  (i)  or  in  others 
involving  the  expression  i  +20  cos  #+a2,  a  development  in  con- 
verging series  is  readily  obtained.  For,  denoting  such  a  value 
by  a',  we  have 

i  +20!  cos  x+a'2  =  a'2(i  +20,  cos  x+a2), 

*  See  Fig.  61,  p.  35  3<  for  the  graph  of  such  a  function. 


336  DEFINITE  INTEGRALS.  [Art.  313. 

in  which  a  stands  for  the  reciprocal  of  a'  and  is  therefore    less 
than  unity.     In  this  way,  equation  (i)  becomes 


dx  i     F          [  sin  x     sin  2X    sin  3^ 


n 
J' 


and  equations  (2)  and  (3)  become 

f*  dx  7T 

J0  i  +2d'  cosx+a'2     a'2  — i 
and 

p          dx  i    PTT  1 

—. —      72  =  ~72 -~2Cot~la  '    > 

J0i+2a  cosx+a       a    —  i  L2  J 

the  values  of  both  integrals  being  essentially  positive. 

When    the   more   general   expression    A  +B  cos  x   (in    which 
A>B  numerically)  occurs,  we  make,  as  proposed  in  Art.  310, 

A  +B  cos  x  =  m(i  +20.  cos  x+a2) 
by  putting 

A=m(i+a2),         B  =  . 


Eliminating  m  and  solving  the  quadratic  for  a,  we  have  recip- 
rocal roots  which  we  may  write  in  the  form 


A- 


whence  also 

A+ 


m=- 


*  In  these  equations,  A  is  regarded  as  positive  (which  does  not  affect  the  gener- 
ality of  the  results),  and  the  radical  is  by  hypothesis  real.  Thus  a  denotes  that 
root  which  is  numerically  less  than  unity,  and  its  sign  is  that  of  B. 


§  XX.]  INTEGRALS  IN   MULTIPLE-ANGLE  SERIES.  337 

For  example,  we  have,  from  equation  (i),  Art.  312, 
dx  i 


T.X- 


on  substituting  the  values  of  m  and  a  above. 

This  integral  has  already  been  expressed  in  finite  form,  thus, 
see  formula  G,  p.  41 : 

r*      dx  2  r  IA-B 

"TTTT  tan 


,A+Bcosx 


U 


The  inverse  tangent  in  this  expression  is  therefore  a  quasi- periodic 
function,  and  substituting  the  values  of  A  and  B  in  terms  of  a 
and  b  we  find 

r  i  —  a        x~\     x     r  a2   .  a3    .  "| 

tan"1    tan  —    = \asmx sin  2X+—  sin  T>X—  . . .     . 

[_i+a        2_\     2     [_  2  3  J 

314.  From  equation  (5),  Art.  311,  we  have  by  integration 

f*  r~  a2  a3  •"] 

Iog(i  +  2acos^+a2)^  =  2    asin^ — ^sin  2^+-^  sin  3^-. . .      (i) 

when  a<i.     Again,  since,  in  the  notation  of  the  preceding  article, 

log(^4  +  B  cos  x)  =log  w  +log(i  +2d  cos  ^  +a2), 
we  have 

f* 

log(4  ^h-S  cos  »)<fof 

^  +  i/M2-5»)       r              fl2  e  -i 

=  iclog 1-2  I  a  sin  A; ;;  sm  2*  +  . . .     . 


338  DEFINITE  INTEGRALS.  [Art.  314. 

Thus  the  integral  takes  in  general  a  quasi-periodic  form  and 
is  a  pure  periodic  function  only  when  it  can  be  put  in  the  form  (i) 
with  a<i.*  The  case  in  which  a>i  in  the  integral  of  equation 
(i)  belongs  to  the  general  case,  the  corresponding  value  of  m 
being  a2;  thus,  when  x=n,  we  have 


f* 

log( 

Jo 


cos  x+ax  =  2xoga    or    o, 


according  as  a  is  greater  or  less  than  unity. 

315.  The  series  in    Art.  311  lead  to  the  immediate  evalua- 
tion of  definite  integrals  of  a  certain  form.     For  example,  if 


f* 

U=     cos  rx  log(i  +  2a  cos  x  +a2)dx, 

J  o 

where  a<i  and  r  is  a  positive  integer,  we  have  by  equation  (5) 

f* 

U  =  2\  (a  cos  x  cos  rx  —  £a2cos  2X  cos  rx  +^a3cos  3^  cos  rx  —  .  .  .)dx. 

J  o 

By  Art.  33,  every  term  of  the  integral  vanishes  except  the  rih 
term.     Thus  we  have 

(jr  ft  /  _  Q\T  ft 

U=  —  2(  —  i)r—     co$2rxdx=  --  •  —  . 
r)0  r 

Compare  Art.  321. 


*  The   necessary  condition  is  ^4  =  i+JB2,  with  the  further  restriction  that 
B  shall  not  exceed  2. 


§  XX.]  EXAMPLES  339 

Examples  XX. 
t* 

i.  Show  that      log  cot  x  dx  vanishes  when  X=\TI  and  has  a  max- 

J  o 

imum  value  when  X=\K.     Sketch  the  forms  of  the  graphs  of  this 

r*  r* 

integral   and   those  of      log  sin  x  dx  and  |    log  cos  x  dx.     See    Arts. 

Jo  Jo 

293  and  287. 

Evaluate  the  integrals: 


'•I 


3 

00  X7  dx  7T 


6. 


9  sin  20° 

£E 
3' 


27T 


>+a3)t/#'  3 

8.  Show  that,  when  «>i, 


7T  7T 

•  =  —  sec  — . 

1       2H  2H 


f°°  dx 

g.  Evaluate       log(i  +  2a  cos#+o2)  2       2  when   c<i,    employing 


the  result  of  Ex.  XIX.,  23.  —  log  (i  +  ae~c). 


10.  Evaluate 


-~- 
J0c' 


c 


—r~\  —  v 
cos  bx+a£ 


340  DEFINITE  INTEGRALS.  [Ex.  XX- 

ii.  Show  that,  if  b  and  c  are  positive, 

log(62±  2bc  cos  x+c2)dx=  27:  log  k, 

J  o 

where  k  is  the  greater  of  the  quantities  b  and  c.    Hence  show  by  differ- 
entiation that 

b  +  c  cos  x 


£ 


:dx=~r      or      o 


cos^+c2         b 
according  as  b  >  c  or  6  <  c. 

I""       #  sin  x  dx 
12.  Evaluate        — ; —         — ; — 5  when  c<i  and  when  c>i. 

Jo  I 


cos 


TC   .  I  7T    ,  a 

—  log ;    —  log . 

a         i  — a      a         a— i 


f  *        cos  rx  dx 

IT..  Evaluate        -  ;  —  5-  where  r  is  a  positive  integer  and 
J0  i  —  20  cos  x-\-a£ 


i - 


14.  Show  that,  when  a<i, 


2  f  « 
(i-a2) 

J0 


<fo  Tr2       /      a3     a5    a7 

—  5=  --+4(a+-^+-2  +  -K 
cos  ^+aj     2        \      3      5      7 


15.  Derive  the  expansion 

T*  a  sin  x  I       a3    a5  \ 

tan"1—  -    —  dx=2[a+^+-~+  .  .  .  I; 
J0  i  +  a  cos  x  \      3      5  / 

and  thence  show  that 


I     .     I 


7T2 


Compare  Diff.  Calc.,  Art.  238. 

f"  sin  x  sin  rx  dx 

1 6.  Evaluate         ; — ,  when  a<i  and  when  a >i,  r  being 

J0  i  —  20  cos  #+ or 


Tta1 
a  positive  integer. 


§  XX.]  EXAMPLES.  341 

17.  Show  that,  when  a<  i, 

f  °°     x  sin  x  dx  7ie~c 


J0  cz+x2  1  +  20  cosx+a2     2(1  +  ae  c)' 
18.  Derive  the  expansion 

a  sin  x 


I    ,  a2  ,  a3  ,        \ 
dx=7i(a-\  —  O+-Q+  .  .  .  I 
\       2      3  / 


i  +  a  cos 

when  a<  i,  and  verify  the  result  when  a=  i. 
19.  Derive  the  expansion,  when  a<  i, 

[*  '       I  Cfl  d^  ( 

rvlog(i  —  2acosx+a2')dx=4(a-\  —  ^-\  —  §+- 
Jo  \      3      5      .7 

and  thence  deduce 


6  log  sin  6  dd=  —•  ,      •  ^  ,    „  i    •> 

8       2  v    33  s3  r 

20.  Evaluate  the  integral         — ^-dx,  where 

Jo     -""I1       x) 

7i  cot  pn. 
Expand  and  see  Diff.  Calc.,  Ex.  XX I II.,  20. 

XXI. 

Functions  expressed  in  Multiple-angle  Series. 

316.  The  infinite  series  involving  cosines  of  the  multiples  of 
an  angle,  namely, 

C+Ai  cos  0+A2  cos  26  +AS  cos  3#+  . . . , 

is  convergent  for  all  values  of  6,  provided  the  absolute  values 
of  the  coefficients  -form  a  convergent  series.      The  series  then 


342  DEFINITE  INTEGRALS.  [Art.  316. 

expresses  an  even  periodic  function  of  6.  Under  similar  circum- 
stances 

BI  sin  0  +B2  sin  26  +B3  sin  3^  +  ... 

expresses  an  uneven  periodic  function  of  6.  In  either  case,  the 
period  is  27r,  and  the  values  of  the  function  corresponding  to 
values  of  6  between  o  and  x  determine  the  values  of  the  function 
for  all  values  of  6. 

So  also,  if  we  put  6=nx/l,  the  series  express  even  and  uneven 
periodic  functions  of  x,  in  which  x  =  l  corresponds  to  $  =  TT,  so 
that  the  values  of  the  function  for  values  of  x  within  the  half- 
period  from  o  to  /  determine  all  the  values  of  the  function. 

It  is  frequently  desirable,  especially  in  the  physical  applica- 
tions of  mathematics,  to  express  a  given  function  f(x)  in  one  of 
these  forms,  and  it  was  found  by  Fourier  that,  notwithstanding 
the  periodic  character  of  the  series,  it  is  possible  so  to  determine 
the  coefficients  in  either  series  that  the  sum  of  the  series  shall  for 
a  range  of  values  of  x  from  o  to  I  represent  any  given  function  f(x). 

Fourier  s  Series. 
317.  Let  us  now  assume  that  it  is  possible  to  put 

xx  TTX  TTX 

+  .  .  .  ,       (l) 


for  all  values  of  x  between  o  and  /;  *  the  values  of  the  coefficients 
can  then  be  determined  as  follows  : 

*  The  demonstrations  of  Poisson  and  Lagrange  establish  the  possibility  of 
equation  (i)  in  a  direct  manner.  In  that  of  Lagrange,  the  series  is  at  first  assumed 
to  consist  of  a  limited  number  of  terms.  The  coefficients  (n  in  number)  are  then 
so  determined  that  the  equation  is  satisfied  for  n  equidistant  values  of  x,  sub- 
dividing the  interval  between  o  and  1.  Thus  the  graph  of  the  series  is  made  to 
coincide  with  that  of  f(x)  at  n  points.  Afterward  n  is  made  infinite,  so  that  the 
graphs  coincide  for  an  unlimited  number  of  points. 


§  XXL]  FOURIER'S  SERIES.  343 

We  have  seen  in  Art.  33,  that  if  m  and  n  are  positive  integers, 

f* 
cos  md  cos  nd  dd  =  o     or     JTT, 

Jo 

according  as  m  and  n  are  unequal  or  equal.  It  follows  that, 
putting  xx/l  =  d,  if  equation  (i)  be  multiplied  by  cos  mO  d6,  and 
then  the  integral  of  each  member  be  taken  between  the  limits 
o  and  Tt  for  6,  that  is  between  o  and  /  for  x,  every  term  in  the 
second  member  will  disappear  from  the  result  except  that  in 
which  n  =  m.  Thus  we  have,  when  n  >  o, 


f*f\C    M    /7 'V*  —      A         • 

V^Uo    /*•        ,       U/Jv  -il  ft  • 

/  2 


ry  r  TT^C         *?  r^     /  Lu  \ 

n  =  -r-    /(^)  cos  w  -rdx  =  —    f(  —  )cosn6dd. 

/I  /  7T  I  ~       \  7T  / 

"Jo  'vjo\'"/ 


(2) 


Again,  multiplying  equation  (i)  by  <#?  and  integrating, 

r/ 


whence 

T  r'  , 

f(v\tiv  *  ^'>^ 

W"* 13; 


Using  a  separate  symbol  for  the  current  variable  in  the 
integrals,  the  result  may  be  expressed  thus: 

j    fl  2n~'X'  TtX  f '  TT^ 

/(*)  — T-    f(v)dv  +-T-  2  cos  w  -y    /(v)  cos  »  -r-rfv,        (4) 

^  Jo  •    M=I  *   Jo  * 

which  is  true  for  all  values  of  x  between  o  and  /. 

*  The  absolute  term  C  is  in  form  the  same  as  \A0  where  A0  is  defined  by  equa- 
tion (2),  but  the  case  n  =  o  sometimes  fails  to  be  included  in  the  general  evalu- 
ation of  the  definite  integral. 


344 


DEFINITE  INTEGRALS. 


[Art.  318. 


318.  For    example,    let    us    put   f(x)=x.      Substituting    in 
equation  (2),  we  have 


2/ 


2/ 


d>  cos 

r 


Integrating  by  parts,  we  find 

,M7T  -|«*  fWt 

<£.cos  (/></<£  =  </>  sin  <fr       —      sin  (f>d(f>  = 

Jo  -J  o          J  o 


mi  —  i  =o     or     —2, 


according  as  n  is  even  or  odd.     Therefore  An=o  when  w  is  even, 

4/ 
and  4»=  —  -2~2  wnen  w  is  °dd.     Again,  equation  (3)  gives  C  =  \l. 


Hence,  when  o  <  x  <  I, 


i  THV     i 

+~2  COS  3y  +~2  COS  5         +  .  .  . 


(i) 


This  result  holds  also  for  the  extreme  values  o  and  /;   as  is 
readily  verified  by  means  of  the  equation 

III  7T2 

I  +  ~5  +~o  +~5+    .   .   .     =-5-, 

32     52     72  8' 

proved  in  Diff.  Calc.,  Art.  238. 

The  complete  graph  of  the  second  member  of  equation  (i) 
^incides  with  the  line  y  =  x  for  the 
interval    between    x  =  o  and    x  =  l, 
Since  the    cosine-series    is  an  even     /^\ 
function,  the  portion  between  x=o  \ 

and  x=—l  is   symmetrical  to  this    '  i 

with  respect  to  the  axis  of  y,  and  the 
rest  of  the  graph  is  a  periodic  repeti- 
tion of  these  parts  forming  a  broken  line  as  represented  in  Fig.  57. 


§  XXL]  THE  SERIES  IN    MULTIPLE   SINES.  345 

The  Series  in  Multiple  Sines. 
319.  In  like  manner,  we  may  assume 

TtX  KX  TCX 

f(x)=Bi  sm  -y  +B2  sin  2  -j  +B3  sin  3 -y  +. . .  (i) 

Thenr  since,  by  Art.  33, 

r 

sin  mO  sin  nd  dO  =  o     or    JTT, 

J  o 

according  as  m  and  n  are  unequal  or  equal,  multiplying  equa- 
tion (i)  by  sin  mO  dO  and  integrating  between  o  and  TT,  we  find 

2    Cl  7IX  2    Cx         flO\ 

Bn  =  -r\  f(%)  sin  n—rdx=—     fl—hmntidO;      ,     (2) 

*  Jo  *  ^JoV"/ 

and  the  result  of  substitution  in  equation  (i)  may  be  written 

2  M=0°         TIX  rl  -v 

/(^)=y^sinwy     f(v)sw.n-j-dv.    ...     (3) 

n=  i  ».° 

The  graph  of  the  sine-series  thus  determined  coincides  with 
that  of/<X)  from  A;=O  to  x  =  l,  while  from  ^  =  oto^=— /it  forms 

an  arc  symmetrical  to  this  with  respect 
/\  to  the  origin  as  represented  in  Fig.  58. 

Thus,  unless /(o)=o,  the  sine-series  is 
I        1-X"     a  discontinuous   function   approaching 
different  limits  when  x  approaches  zero 
J(        from  one  side  or  the  other.     The  rest  of 
v  the  graph  is   a   periodic   repetition   of 

these  arcs,  and  thus,  unless  /(/)=  o,  the 
FIG.  58.  .        .       ,. 

series    is    discontinuous    also   at   x  =  l. 

In  each  case,  the  value  of  the  series  at  a  point  of  discontinuity 
is  zero. 


346 


DEFINITE   INTEGRALS. 


[Art.  320. 


320.  As  an  illustration,  let  us  find  the  sine-series  for /(#)=#. 
From  equation  (2),  we  have 


2/  {"         .  ll     [n*        . 

BK  =  ~~7>  I    Q  sin  nO  dd  =   9  9 1    0  si 
T?J  o  »^r  J  o 

Integrating  by  parts, 

f  we  ~~\  nit       fn 

d>  sin  0  d<f>  =  —  <j>  cos  0       +1 

Jo  Jo  Jo 


whence 


cos  <p  a<p=  —HTI  cos  WTT; 


Therefore,  substituting  in  equation  (i), 

2'  /    .      7*30       I      .  7TOC       I      ,  TT^V  \ 

x  =  —  I  sin  —, sin  2  —r  +—  sin  ^  -; —  .  .  .  I .  fi ) 

n\        I      2  I      3         3  /  / 

The  graph  of  this  series  is  not  discontinuous  at  the  origin 
because  the  value  of  /(o)  is'  zero.  In  fact,  in  this  case,  and  when- 
ever f(x]  is  an  uneven  function,  the 
graph  of  the  series  coincides  with  that 
of  f(x]  for  the  whole  interval  between 
x=—  I  and  x  =  l,  but  in  general  exclu- 
sively of  these  Emits.  In  like  manner, 
when/(#)  is  an  even  function,  the  corre- 
sponding cosine-series  will  represent  the 
function  for  all  values  between  x=  -I  and  x=l  and,  in  that  case, 
inclusive  of  the  limiting  values. 

321.  As  a  converse  application  of  the  equations  of  Arts.  317 
and  319,  we  may  remark  that,  when  the  expansion  of  a  function 


FIG.  59. 


5  XXL]  THE  SERIES  IN   MULTIPLE  SINES.  347 

in  sines  or  cosines  of  multiples  of  x  has  been  otherwise  obtained, 
the  known  values  of  the  coefficients  give  the  values  of  the  definite 
integrals  which  occur  in  those  equations.  For  example,  we  .have 
seen  in  Art.  315  that  the  cosine-series  for  log (i  +  2<zcos#+a2)  gives 
the  value  of 


where  r  in  an  integer,  the  process  employed  being  the  same  as  that 
employed  in  Art.  317  to  derive  the  general  expression  for  the 
coefficient  An. 

Again,  putting  0  =  1  in  equation  (5),  Art.  311,  we  derive 

log  COS  \X=  —log  2  +COS  X  —  \  COS  2X+%  COS  ^X  —  .  .  .  , 

which  is  convergent  for  all  values  of  x  up  to  X=K.  From  the 
various  coefficients  we  can  therefore  infer,  when  n  =  o, 

log  cos  \x  dx  =  xC  =  —7t  log  2 

J  o 

(which  agrees  with  Art.  287);  and  also,  when  n  is  an  integer, 

f  * ,  K  A  ~ 

log  cos  \x  •  cos  nx  ax  = — An=—(—i)n — . 

2  2       n  V         '    2U 


Developments  containing  both  Sines  and  Cosines. 

322.  If>  when./(:v)  is  neither  an  even  nor  an  uneven  function, 
.a  development  in  multiple  angles  applicable  from  x=  —  I  to  x  =  l 
be  required,  it  will  be  necessary  to  separate  f(x)  into  its  even  and 
uneven  parts,  for  development,  the  one  in  cosines  and  the  other 


348  DEFINITE  INTEGRALS.  [Art.  322. 

in  sines.      Denoting  these  by  <f>(x)   and  <l>(x)  respectively,  as   in 
Diff.  Calc.,  Art.  216,  they  are 

+/(-*)!     and     ft*) 

so  that/(#)  =  (j>(x)  +  4>(x). 

Then,  because  the  integral  of  an  even  function  between  the 
limits  —  /  and  /  is  double  the  integral  between  o  and  /,  the  coeffi- 
cients for  <f>(x)  and  ^(x)  respectively,  Arts.  317  and  319,  may 
be  written  in  the  form 


I     fl  If1  7TV 

C=—r\     4>(v)dvt     An=-\     <£(V)  cos  n-rdv. 

2l  J  -I  I*  }  -I  l< 

I    fl 

Bn  =  -j\ 

I  J  _; 


TtV 

sin  n—rdv. 


But,  because  the  integrals  between  —  I  and  /  of  the  uneven  functions 

TtV  TlV 

(b(v)  cos  n—r  and  <f>(v)  sin  n-r  vanish,  the  coefficients  may  also 
/  I 

be  written 

if*  if'  nv 

C=^l]  _  Arfdv,     A  „  =  -  J  ^  f(v)  cos  n-jdv, 

I    f*  7CV 

Bn=j)     /(v)  sin  n-jdv. 


Hence 


=~\  _f(v}dv 


j    H=  00 

+7  2  ( cos  n 

b 


7tx[l  TtV  7tx{1  ,  7tV\ 

—r      f(v)  cos  n—  +sm  n—      f(v)  sin  n—  )dv. 
I  J  —i  t  I  j  —i  I  / 


§  XXI.J        DEVELOPMENT  IN  SINES  AND   COSINES  349 

323.  For  example,  to  obtain  a  development  of  emx  true  for 
all  values  of  x  between  —  /  and  /,  we  have 


i 

emvdv  = 


i  tl 
C  =  -T\ 

2.1}  -i 

fi  rw       wIO 

IP  7CU  I   f" 

An  =  —r\     emv  cos  n-rdv  =  -\     e  *  cos  no  d6,, 

I    J  -I  i  ^J-n 

if'  xv      i  r   — 

En  =  -T\     emv  sin  n—  dv  =-\     e*  sin  nd  dd. 

I    J  -I  I  ~J  -x 

To  evaluate  these  integrals,  we  have,  by  Art.  63, 

f  em9 

emd  cos  nd  dd  =  —5  -  ~(m  cos  nd  +  n  sin  nd), 
j  m2  +w2V 

r  e™** 

eme  sin  nd  dd  =  —z  -  z(m  sin  nd  —  n  cos  n6)  ; 

W2+W2V 


whence 


. 


emd  cos 


ew9  smnOd6=-(-i  Y 

j  _K  ' 


-  e 


Hence,  putting  ml/it  for  m,  and  substituting, 


/*^J?//>W/  /?  ^  W/ \  4/J,— /X>WW S>~"  Wll\ 

.     ffil  I  c  t>  i  f i/n.  i  e  e 
/ T \n_                                                           /      _\« 


350  DEFINITE   INTEGRALS.  [Art.  323. 

Therefore 


XX  XX 

cos  —         cos  2-7- 
I  /  / 


\2m2l2     m2l2  +x 


sm  2  sin  2  3  sm 


The  first  line  is  of  course  the  development  of  cosh  mx,  and 
the  second  that  of  sinh  mx  (Biff.  Calc.,  Art.  217).     Thus,  taking 


2 m  sinh  mx  I  i        *  ( —  i  )M  cos  w#\ 

coshwx  =  -  -I — z  +  2 „— — s — ), 

/r  \2W^      j        m2+n2     J 

2  sinh  mx  *  —  ( —  i )"  w  sin  WA; 


sinh  mx  = 


- 
m2+n2 


Discontinuity  of  the  Fourier  Sine-series. 

324.  We  have  seen  in  Art.  319  that  the  Fourier  sine-series 
is  in -general  a  discontinuous  function,  presenting  sudden  changes 
of  value  as  illustrated  by  the  graphs  Figs.  58  and  59.  The  mode 
in  which  an  ordinary  sine-series,  which  represents  a  continuous 
periodic  function,  takes  on  this  character  is  well  illustrated  by 
equation  (7),  Art.  311,  namely, 

a  sin  x  a2  a3 

sm  2X-\ — sin  *x  — .  .  . ,    d) 


i  +a  cos  JP  23 


§  XXL]    DISCONTINUITY  OF  THE  FOURIER  SINE-SERIES.    351 

in  which  a<i,  so  that  the  series  is  convergent  for  all  values  of  x. 
When  x  =  o,  the  value  of  the  series  is  zero,  so  that  the  symbol 
tan  -1  must  then  be  taken  to  mean  the  primary  value  of  the  inverse 
tangent.  Now,  since  the  denominator,  i  +a  cos  x,  cannot  vanish, 
y  cannot  reach  either  of  the  values  fa  or  —  fa,  and  therefore  the 
tan"1  is  for  all  values  of  x  restricted  to  denoting  the  primary 
value  which  lies  between  fa  and  —fa. 

When  a>i,  the  series  is  divergent  and  equation  (i)  ceases 
to  have  any  meaning. 

In  the  intermediate  case,  when  a  =  i,  the  series  remains  con- 
vergent for  all  values  of  x,  and  becomes,  in  fact,  the  Fourier 
sine-series  for  $x,  when  we  take  l  =  it  in  equation  (i),  Art.  320. 
Accordingly,  the  function  y  now  takes  the  form  tan"1  tan  %x, 
of  which  one  value  is  \x.  But  the  series  always  represents 
the  primary  value  of  the  tan"1  (which  is  §x  only  when  x  is  be- 
tween -it  and  TT);  thus  it  becomes  a  discontinuous  function, 
increasing  with  x,  but  dropping  the  value  it  suddenly  whenever 
x  passes  through  an  odd  multiple  of  it. 

325.  The  development  of  the  same  function  y  in  a  series 
convergent  when  a>i  is  readily  obtained  from  Art.  311,  in  which 

i  +aeix 

2iy  =  log  ; -•-. 

&  i  +ae~tx 
For  this  may  be  written 

i    _ . 
a 


i    . 
a 


in  which,  if  a>i,  the  second  term  can   be  developed  as  in  equa- 
tion (6).     Hence,  dividing  by  21,  we  have,  when  a>i, 


352  DEFINITE  INTEGRALS.  [Art.  325. 


a  sin  x  /sin  x    sin  2X     sin  T.X 


i  -fa  cos  #_  \    a 


\ 

-...     .      (2) 
/ 


The  series  in  this  equation  is  convergent,  and  is  an  ordinary 
periodic  function,  so  that  y  is,  in  this  case,  a  quasi-periodic  func- 
tion. 

If  we  now  make  a=i  in  equation  (2),  the  value  of  y  becomes 
x  diminished  by  the  Fourier  series  for  $x,  and  is  again  equal  to 
$x  when  x  is  between  —  it  and  TT;  but  as  x  increases,  this  ex- 
pression for  y  receives  a  sudden  increase  of  TT  in  value  when  x 
passes  through  an  odd  multiple  of  TT. 


Geometrical  Illustration. 

326.  The  function  y,  developed  in  equations  (i)  and  (2)  above, 
admits  of  a  simple  geometrical  construction.  For,  if,  in  Fig.  60, 
we  take  BA  =i  and  AP.=  a,  and  denote  the  arcual  measure  of 
the  angle  PAC  by  x,  that  of  the  angle 
PBC  will  represent  y,  where 
a  sin  x 


i  +acosx 

As  x  increases  indefinitely,  P  describes 
a  circle;  and,  supposing  <z<i,  as  in  the 
diagram,  y  is  a  periodic  function  return-  FlG-  6o- 

ing  to  a  given  initial,  value  when  x  is  increased  by  2-n.  The  figure 
represents  the  "slit-bar"  mechanism,  in  which,  when  AP  is  nearly 
equal  to  AB,  the  bar  BP  vibrates  with  a  "quick-return  motion," 
whereby  a  quantity  nearly  equal  to  n  is  subtracted  from  the 
angle  y  whenever  P  passes  through  a  certain  small  arc  in  the 
neighborhood  of  D. 

Now  let  a  =  i,  so  that   B  falls  on  the  circumference  of  the 
circle;  then  we  have,  for  values  of  x  less  than  it,  y  =  ^x;  and  we 


§  XXL] 


GEOMETRICAL  ILLUSTRATION. 


353 


may  take  this  as  universally  true,  making  the  bar  BP  rotate 
uniformly  at  half  the  rate  of  A  P.  But  the  series  in  equation  (i) 
continues  to  represent  the  primary  value  of  the  position  angle  y, 
which,  as  x  increases,  is  subject  to  a  periodic  dropping  of  the 
value  TI  corresponding  to  the  quick  return. 

327.  On  the  other  hand,  when  a>i,  the  point  B  falls  within 
the  circle,  and  the  value  of  y  is  represented  by  equation  (2). 
The  bar  BP  now  makes  complete  revolutions,  and  when  a  is  near 
to  unity,  swings  very  rapidly  through  a  half-revolution  in  the  posi- 
tive direction.  Hence, 
when  the  limiting  case  is 
reached  by  diminishing 
a  to  unity,  the  value  of  y 
increases  uniformly  at 
half  the  rate  of  x,  but 
receives  a  sudden  increase 
of  TT  in  value  whenever  x 
passes  through  an  odd 
multiple  of  n. 

The  graphs  of  the 
function  /for  two  recip- 
rocal values  of  a,  near  to 
unity,  are  given  in  Fig.  61. 
The  zigzag  lines,  bisected 

respectively  by  y  =  o  and  y  =  x,  are  the  limiting  forms  to  which 
the  graph  approaches  as  a  approaches  unity  from  the  one  side 
or  the  other. 


FIG.  61. 


Differentiation  of  Multiple- angle  Series. 

328.  The  differentiation  of  a  sine-series  or  a  cosine-series  for 
f(x)  gives  a  cosine-series  or  a  sine-series  for  f(x).  When  the 
series  is  an  ordinary  continuous  function,  and  therefore  either 


354  DEFINITE  INTEGRALS.  [Art.  328. 

an  uneven  or  an  even  one,  this  is  in  accordance  with  the  fact 
that  the  derivative  of  an  uneven  function  is  an  even  one,  and 
vice  versa.  Moreover,  the  sine-series  has  the  property  of  the  un- 
even function  that  f(o)  =  o. 

But  when  the  series  is  the  Fourier 'sine-series  for  an  arbitrary 
function /(#),  we  have  not  generally  f(o)=o,  although  the  sine- 
series  as  assumed  in  Art.  316  vanishes  with  x.  This  fact  imposes 
discontinuity  upon  the  series,  /(o)  being  not  the  value  of  the 
series  when  x=o,  but  the  limit  to  which  the  value  of  the  series 
approaches  when  x  approaches  to  zero  from  the  positive  side. 

329.  The  derivative  of  the  Fourier  cosine-series  for  f(x)  will 
in  fact  give  the  sine-series  for  f(x},  true  between  the  same  limits. 
But  it  is  to  be  noticed  that  the  derivative  of  the  sine-series  for 
f(x)  cannot  contain  the  term  C,  independent  of  x;  therefore  it 
cannot  give  the  cosine-series  for  f(x)  when  the  latter  properly 
contains  this  term.     The  failure  of  differentiation  in  these  cases 
is  due  to  the  fact  that  a  divergent  series  is  produced. 

For  example,  putting  l=n  in  equation  (i),  Art.  320, 

#  =  2 (sin  x  —  \  sin  2X+\  sin  yx  —  .  .  .  ),    .     .     .     (i) 
of  which  the  derivative  is 

i  =2(cos  x— cos  2X+CO5  $x  +  . . .  ), 

which  is  inadmissible,  because  the  series  is  divergent. 

330.  Let  us  now,  in  the  general  theorem,  put  l  =  n  (which  is 
the  same  thing  as  putting  x  in  place  of  its  multiple  nx/l,  and 
therefore  involves  no  loss  of  generality).     Then>  in  the  cosine- 
series  for/(#),  Art.  317,  the  coefficient  of  cos  nx  is 

2  f" 
An=-\  f(x}  cos  nx  dx. 


§  XXL]        DERIVATIVES  OF  MULTIPLE-ANGLE  SERIES.  355 

Integrating  by  parts,  this  becomes 

2  [•/(*)  sin  nx    if,,,,.          ,  ~|* 

An=-\  \  f  (x)  srn  nx  ax  \ 

nl         n  nj  J0 

2  r                             i 
= /  f(x)  sin  nx  dx= Bn) 

Tlflj     Q  .  ft 

where  B'n  stands  for  the  coefficient  of  sin  nx  in  the  sine-series 
for  f'(x)  as  found  by  the  method  of  Art.  319.  Thus  B'n=  —  nAn; 
but  this  is  precisely  the  coefficient  of  sin  nx  in  the  derivative  of 
the  cosine-series  forf(x).  Thus  the  derivative  series  is  the  same 
as  the  result  of  developing  f'(x)  in  a  Fourier  sine-series  by  the 
general  theorem. 

331.  On  the  other  hand,  the  coefficient  in  the  sine-series  is 

2  f* 

Bn  =  —  \    f(x)  sin  nx  dx, 
n  J  0 

and,  integrating  by  parts, 

2  r-/(*)  cos  «# 


which  shows  that  we  shall  havev.4J,  =  «.Bw,  0^/3;  ?w  Cfl^e  /"(o)=o 
and  f(n)=o.     In  this  case  only,  therefore,  and  not  in  general,* 

*  The  expression  for  Bn  above  shows  that  the  sine-series  for  f(x)  can  be  sepa- 
rated into  three  parts  as  follows: 


f(x)=A'\  sin  x-\-\Ai  sin  zx-\-  $A3  sin 
+  —  —  [sin  x+  i  sin  2X-\-  $  sin 


7T 
2 


[sin  *— J  sin  2X+  J  sin  3^;—  .  . .  ]. 


356  DEFINITE  INTEGRALS.  [Art.  331. 

will  the  sine-series  admit  of  a  convergent  derivative.  It  is  to  be 
noticed  that  this  is  precisely  the  case  in  which  the  equation  of  f(x) 
to  the  sine-series  is  true  for  the  limits  inclusive,  in  which  case  the 
graph  of  the  series  is  a  continuous  curve  composed  of  arcs  which 
meet  each  other  with  a  common  tangent.  For  example,  X(TZ  —  x) 
is  such  a  function,  and  the  corresponding  sine-series  (with  a 
graph  consisting  of  connected  parabolic  arcs)  will  be  found  to 
give  a  convergent  cosine-series  for  the  derivative  T:  —  2X.  The 
result  is,  in  fact,  equivalent  to  equation  (i),  Art.  318. 


Integration  of  Multiple-angle  Series. 

332.  Conversely,  if  f(x}  is  a  given  function  of  which  the 
development  in  a  cosine-series  contains  no  absolute  term,  that  is, 

f* 

if  C'=o,  we  shall  have      f'(x}dx=o,  whence  /(/:)  =/(o).     The 

•  o 

constant  of  integration  can  be  so  taken  that  the  integral  f(x) 
shall  vanish  when  x=o,  and  we  shall  then  have  also/(-)=o. 

The  sum  of  the  series  in  the  third  line  is,  by  equation  (i),  Art.  329,  equal  to  \x, 
and  that  in  the  second  line  is  the  result  of  substituting  n  —  x  for  x  in  the  same 
equation;  hence  the  last  two  lines  are  equivalent  to 


and  we  have 

~  (i) 


Since  C'=—  I     f'(x)  dx  =  ^—^- ,  the  derivative  of  this  equation  is 


i    f* 

H?J  /'w 


cos  2x+A3  cos  3*+  .  .  .  , 

as  given  directly  by  Fourier's  method.  Thus  in  equation  (i)  the  part  of  the 
?:ne-series  which  produces  a  divergent  derivative  has  been  already  evaluated  and 
the  remainder  admits  of  a  convergent  derivative. 


§  XXL]      INTEGRATION   OF  MULTIPLE-ANGLE  SERIES.          357 

The  sine-series  corresponding  to  f(x)  with  the  constant  thus 
determined  will,  as  shown  above,  have  the  property  of  actually 
approaching  zero  when  x  =  o,  and  also  when  x  =  ~. 

In  the  case  of  any  given  value  of  f'(x),  we  obtain  a  function 
of  the  required  kind  by  simply  transposing  '  C'  in  the  develop- 
ment in  cosines.  For  example,  from  equation  (i),  Art.  318,  we 
have 


cos  3#  +  -^  cos  5#+  .  .  .  I;    .     (i) 


whence,  integrating, 

8/.          i  i  \ 

•  —  xz  =  —  IsmjcH — 5  sin  T.X+-^  sin  zx+  .  .  .  /,       .     (2) 
it\  33  53  / 


xx- 

3°  5J 

the  constant  of  integration  being  zero,  so  that  the  first  member 
vanishes  when  x  =  o,  and  it  is  found  to  vanish  also  when  x  =  n. 

333.  In  integrating  a  sine-series  the  direct  determination  of 
the  constant  is  not  so  readily  made,  but  by  proceeding  to  the 
next  integration  we  can  determine  both  constants  by  the  double 
condition  employed  above.  For  example,  integrating  the  equation 
obtained  above,  and  multiplying  by  —3,  we  have 

3?r  24/  i  i 

x3— —x2+C  =  —  (cos  x-\ — -A  cos  ix H — -.  cos 

2  x\  34  S 


and  integrating  again, 


--  — 
42 


—  (sin  ^+-=  sin  3^+3=  sin  5*+  .  .  .), 

ic  \  35  55 


in  which,  the  two  conditions  that  the  first  member  must  vanish 
when  x  =  o  and  also  when  x  =  x  give  C'=o  and  C  =  j7r3. 


358  DEFINITE  INTEGRALS.  [Art.  333. 


Hence  the  successive 

integrals  are 

T.7Z      ,        7T3 

£•3  _±L_   x;2  _j  

2           4 

24  / 

~7T\ 

i 

cos^+^a^ 

i 
H  —  T  cos  ^ 
54 

....), 

(3) 

06  / 

f  .          i 

i 

\ 

X*-2XX*+a*X 

'  =  -\ 

sin  x  -i  —  ~  sin  33 
^            3 

"+-5  sin  5^; 
o 

+  ...J. 

(4) 

334.  We  have  seen  that  the  sine-series  obtained  directly  for 
a  given  function  is  in  general  discontinuous,  and  represents  the 
function  for  values  between  but  exclusive  of  the  limiting  values 
of  the  variable.  But  the  preceding  articles  show  that  the  expression 
for  the  function  found  by  integrating  the  derivative  as  a  cosine- 
series  gives  its  true  value  inclusive  of  the  limits.  The  value  of 
the  sine-series  thus  found  differs  from  the  given  function  by  a 
certain  linear  function. 

As  an  illustration,  if  we  develop  emx  in  a  sine-series  for  values 
of  x  between  o  and  it  (see  the  formulas  of  Art.  323),  we  have 


Bn  =  —  I    ^'"r  sin  w#  Jjc  =         2       2Ji  -  (  - 1  }nem*\ 

giving  the  development 

2/i+em*  '.  i—em*  .  i+em*  .  \ 

Again,  developing  emx  in  a  cosine-series,  we  have 

i  f  *  em*  —  i 

C=-     emxdx=_ 

71 J0  W7T 

and 

^4«  =  -     ew*cos  nxdx=  ——5-. — ^\(  —  i}nem*  —  i  1. 


§  XXL]      INTEGRATION   OF  MULTIPLE- ANGLE  SERIES.         359 

giving  the  development 
emx=e- I     2m 


71    W+I2' 

i—  emn 


COS2X-\ » «>  COS  3#  +  •  •  •    )         (2) 

/i/fiZ     I     **2  *J  /  \     / 


Integrating  and  multiplying  by  m, 


...), 


X  +  I 1  """a""] — 2sin  aH — r~?~^ — ^rsin  2#+ . . .),    (3) 

Tt     \W2+I2  2(w2+22)  / 

the  constant  of  integration  being  determined  by  either  of    the 
conditions  that  the  series  vanishes  when  x=o  and  when  x=n. 
335.  The  coefficient  of  sin  nx  in  the  last  series  is 


Tt      n(m2+n2)  ' 

and  subtracting  this  from  Bn,  the  coefficient  in  equation  (i),  we 
have 

2[l-(-l)   e      I       2+m2.=  2_  rj_/      jNngWKl. 

7tn(m*  +nz)  mi 

therefore  subtracting  equation  (3)  from  equation  (i)  we  find 

emn  —  i             2  M  =  0°  i  —  (  — i)MfWjr   . 
^  +  i=—  ^   • — sin  nx. 

7i  7i  n 

n  =  i 

This  equation  is  readily  verified  when  we  substitute  in  the  first 
member  the  values  of  x  and  of  unity  in  the  form  of  sine-series,  viz. 

#  =  2(sin  x  —  Jsin  2^+^  sin  3^—  .  .  .  ),     .     .     .     (i) 
see  Art.  329,  and 


i  =  -[sin  x+%  sin  3^+^  sin  $x+  .  .  .  ],     .     .     .     (2) 

71 


see  Art.  336. 


360  •  DEFINITE  INTEGRALS.  [Art.  335. 

Thus  the  linear  function  in  equation  (3)  forms,  as  it  were,  that 
part  of  the  function  emx  which  requires  for  its  development  in  sines 
a  discontinuous  series.  When  the  complete  graph  of  the  second 
member  of  equation  (3)  is  drawn,  the  linear  function  represents 
the  straight,  line  which  passes  through  the  extremities  of  the  arc 
of  y  =  emx  with  which  the  graph  coincides  between  #  =  o  and  x  =  n. 

Series  obtained  by  Transformation. 

336.  Denoting  by  x  the  angle  whose  multiples  appear  in 
the  series  (which  was  denoted  by  6  in  Art.  316),  the  limits  of 
applicability  of  the  series  derived  directly  by  the  methods  of  Arts. 
317  and  319  are  x=o  and  X  =  TZ;  but  by  transformation  of  such 
series,  others  with  different  limits  may  be  derived.  We  notice 
in  the  first  place,  however,  that  the  substitution  of  TI—X  for  x 
does  not  change  the  limits.  Its  effect,  moreover,  is  merely  to 
change  the  sign  of  alternate  terms,  namely  those  containing 
odd  multiples  of  x  in  the  case  of  a  cosine-series,  and  those  con- 
taining even  multiples  in  the  case  of  a  sine-series.  It  follows 
that  the  sum  of  the  given  series  and  this  transformation,  and 
likewise  their  difference,  will  contain  either  odd  multiples  only  or 
even  multiples  only  of  x. 

For  example,  equation  (i)  of  the  preceding  article  gives 


TT  —  #  =  2(sin  x  +  %  sin  2*+^  sin  3*+  ...),.     .     .     (3) 
and  the  sum  is 

7r  =  4(sin  x+%  sin  3^+^  sin  53:+  .  .  .  ),    ...     (4) 


which  is  equation  (2)  employed  above.     Again,  the  difference  of' 
the  series  for  x  and  for  TI  —  X  is 


sin  4#  +  &  sin  6x  +  ...),.     .     (5) 
which  contains  only  even  multiples  of  x. 


§  XXI.]          SERIES  OBTAINED  BY  TRANSFORMATION.  361 

337.  A  series  -containing  even  multiples  only  of  x  may  be 
transformed  by  putting  (j>  =  2x;  and  then,  since  the  limits  for  x 
are  o  and  TT,  those  for  (f>  are  o  and  2x.  For  example,  equation  (5) 
above  thus  becomes 

TT  —  <£  =  2(sin  <£+i  sin  2(f>+^  sin 

which  is  identical  with  equation  (3),  and  shows  that  that  equation 
is  true  up  to  x  =  2x.  This  might  also  have  been  inferred  from 
the  fact  that  equation  (i)  is  true  between  the  limits  —  n  and  TT. 
If  a  multiple-angle  series  whose  limits  are  o  and  TT  be  trans- 
formed by  putting  \K  —  x  for  x,  the  limits  for  the  new  variable  are 
—  \n  and  +  |TT.  In  general;  the  transformed  series  would  con- 
tain both  sines  and  cosines;  but,  if  odd  multiples  only  occur,  a 
sine-series  will  thus  give  rise  to  a  cosine-series,  and  vice  versa. 
For  example,  equation  (4)  above  becomes 

7T 

—  =cos  x  —  %  cos  3#+i  cos  $x  —  .  .  . , 

which  is  true  for  values  of  x  between  the  limits  ±^?r,  exclusive 
of  both  limits. 


Functions  with  Arbitrary  Discontinuities. 

338.  The  restriction  of  the  validity  of  equations  (4),  Art.  317, 
and  (3),  Art.  319,  to  values  of  x  between  o  and  I  presents  itself 
as  a  natural  consequence  of  the  fact  that  the  definite  integrals 
in  the  second  member  depend  for  their  values  solely  upon  the 
values  of  the  function  for  values  of  x  between  the  limits  of  inte- 
gration. 

In  fact,  it  is  found  that  the  equations  are  true  when  the  values 
of  the  function  for  values  of  x  between  the  limits  are  defined  in 


362 


DEFINITE  INTEGRALS. 


[Art.  338. 


L    X 


any  manner  whatever,  provided  only  there  is  no  ambiguity  about 
the  values  of  the  definite  integrals.  Thus  f(x)  may  be  defined 

by  one  functional  expre-sion,  say 
fi(x),  from  x  =  o  to  x  =  a,  by  an- 
other expression,/^),  from  x  =  a  to 
x=b;  and  so  on  to  x=L  It  is  not 
even  necessary  that  the  values  of. 
fi  (a)  and  f^(a)  shall  be  the  same. 
FlG-  62-  In  other  words,  the  graph  of  y  =f(x) 

may  consist,  as  in  Fig.  62,  of  any  disconnected  arcs,  of  which  the 
projections  upon  the  axis  of  x  cover  without  overlapping  the 
range  from  x  =  o  to  x  =  l.  The  values  of  the  definite  integrals 
are  now  found  by  separate  integrations  of  the  functions  f\(x], 
fz(x)  etc.,  each  over  its  proper  range.  In  particular,  that  occurring 
in  the  value  of  C,  equation  (3),  Art.  317,  namely 


=  lj 


represents  the  total  area  between  the  axis  of  x,  the  arcs,  and 
their  ordinates,  due  regard  being  paid  to  algebraic  sign.  Thus 
the  absolute  term  is  the  mean  value  of  the  ordinate  between  *=o 
and  x=l. 

It  is  a  noteworthy  fact,  which  we  here  state  without  proof, 
that  the  value  of  either  series  corresponding  to  a  value  x=a 
at  which  discontinuity  occurs  is  $\fi(a)+f2(a)},  that  is,  the 
arithmetical  mean  between  the  two  values  of  the  function. 

339.  As  an  illustration,  let  us  construct  a  cosine-series  in 
which  fi(x)=x  from  *=o  to  x  =  %l,  and/2(^)=o  for  all  values 
of  x  between  \l  and  /.  We  shall  now  have  in  equations  (3)  and  (2), 
Art.  317,  C  =  f/,  and 


TtX 


2/ 


- 

6  cos 


2/ 


cos 


§  XXL]  FUNCTIONS  WITH  ARBITRARY  DISCONTINUITIES.    363 
Integrating  by  parts, 

«*  nn         nn_ 

<j>  cos  0  d<j)  =  <£  sin  <j>  sin  <f>  d(j> 

Jo  -Jo          J  o 


7t    .    nx  mi 

=  w—  sin Kos 1. 

22  2 


The  successive  values  of  this  integral  for  n  =  i,  2,  3,  etc.,  are 

x  37r  5?r 

--  1,  -2,   ---  1,  o,  --  1,  -2,  etc. 

2  22 

Hence,  substituting  in  the  general  equation,  we  have 

/       /  /        Ttx     i  xx     i  7r#  \ 

/<X>  =  0-  +~  I  cos  T  ~7  cos  3  T  +7COS  5  T  ""••'/ 

oTrxfr^  *5  ' 

2l  i          TtX        I  XX        I  7T#  \ 

--5\«>s  y  +-2  cos  3  y  +-2  cos  5  y  +.  .  .) 

If  TtX         I  TlX         I  TTiC 

--    cos  2  --H  —  cos  6  -+-;  ,cos  10--+.  . 


It  is  readily  verified,  by  means  of  Gregory's  series  for  \n  and 
the  series  quoted  in  Art.  318,  that  x=o  and  x  =  l  each  give  to 
the  second  member  the  value  zero.  Again,  if  we  put  x  =  %l,  the 
second  member  becomes 

/      //        i       i  \      / 


and  this  is  the  arithmetical  mean  between  /i(|/)  =  J/  an 
in  accordance  with  the  last  paragraph  of  Art.  338. 


364  DEFINITE  INTEGRALS.  [Art.  340. 

Formula  Involving  both  Sines  and  Cosines. 

34-0.  If,  in  the  half -sum  of  the  formulae  of  Arts.  317  and  319, 
the  factors  sin  nxx/l  and  cos  nxx/l  be  placed  under  the  integral 
signs,  the  result  may  be  written 

I  Cl  I    "  =  °°  f7  7T 

f(x)=  —  \f(v')dv+—  ^   \  f(v)  cos  n  j(v—  x)  dv.     .     (i) 

2/Jo  *  n-J°  l 

In  like  manner,  the  formula  of  Art.  322  may  be  written 

If  I    n          f  TC 

f(x}  =  —j\  f(v)dv  +  —  ^      f(v]  cos  n  —  (v— x]  dv*    .     (2) 

2/J  -i  I  I 

J      I  M  = r  J      * 

Let  Wi=/r/7,  U2  =  27t/l,  etc.,  so  that  un  =  mt/l  and  the  series 
of  w's  have  the  constant  difference  Au  =  n/l.  Equation  (2)  may 
then  be  written 

f(x)  =— :    f(v)  dv-\ —  2      cos  un(v—x]f(v}  dv  Au. 
zlj-i  *»-*)-* 

If,  in  this  equation,  /  increases  without  limit,  so  that  Au  de- 
creases without  limit,  the  summation  with  respect  to  n  will  be 
replaced  by  integration,  the  limits  for  u  being  zero  and  infinity. 

r 

Therefore,  if  f(x}  is  a  function  such  that         f(v}dv  has  a  finite 

J    —00 

value,  we  shall  have 

f(x}  =  -  cos  u(v — x}f(v}dv  du. 

71  J   0   J  _oo 

This  is  known  as  Fourier's  double-integral  theorem. 


*  Poisson's  method  consists  in  a  direct  demonstration  of  this  equation.     See 
Todhunter's  Int.  Calc.,  p.  298. 


§  XXI.]  EXAMPLES.  365 


Examples  XXI. 

i.  Develop  x2  in  a  cosine-series,  taking  /=-;   and  verify  for  .Y=O 
ind  for  X=TI. 


2.  Find  the  sine-series  for /(:*;)  =  i  from  x=o  to  x=l,  and  show 
that  the  result  is  the  derivative  of  equation  (i),  Art.  318. 

4  /  .    xx      i  TIX      i  xx  \ 

i  =  —  I  sin  —  H —  sm  3— H —  sin  5-y+  •  •  •  )• 

\  O  3  / 

3.  Develop  x(l— x}  in  a  series  of  sines. 

8/2/  .    Ttx      i  Ttx      i  nx 

1  T+Ii  sm  3T  +  3a  sm  5—  +  . 


4.  Expand  cos  $6  in  a  series  of  cosines  taking  /=-,  and  show  that 
the  result  is  numerically  true  for  all  values  of  0. 

„  _  4  /  i      cos  6     cos  26     cos  3# 

A2       i-3         3-5          5-7 

5.  Expand  cos  %6  in  a  series  of  sines,  and  show  by  its  graph  that 
the  result  is  true  from  6=0  to  6—271  exclusive  of  these  limiting  values. 

,„     8/sin#     2  sin  26     3  sin  ?0 

cos  i/7=  —  I 1 —  +-         — +... 

x\  i-3          3-5  5-7 

6.  From  Example  4  derive  the  numerical  series 

— i         i i_ 

T    •    •    •    , 

8      1.3     5.7     9.11 
and  verify  by  means  of  Gregory's  series. 

7.  Show  that  the  result  in  Example  i  may  be  derived  by  integrating 
the  expression  for  x  in  sines,  and  determine  the  constant  together  with 
the  result  of  the  next  integration. 

/  •           J     •          ,   x  \ 

xa—7t^x=  —  121  sin  x 5  sin  2X+—^  sin  -ix—  ...  I 

\  23  33  / 


366  DEFINITE  INTEGRALS.  [Ex.  XXI. 

8.  Find  the  result  of  two  more  successive  integrations,  see  Example  7. 

X4  —  2X2X2  +  —  7T4=48(  COS  X  --  COS  2X+~  COS  ^—  .  .  . 


=72o(  sin  x  --  ^sin  2x+~5  sin 

\  <J 

9.  Putting   Sn=i+—  +  —  +  ...as   in    Diff.    Calc.,  Art.    239,  we 

O 

have   also  i  ---  1  ---  ...=i 


(i  --   -l^w.     Derive,  by  putting  x=x 

\  2  / 

\  / 


and  x=o  in  Examples  i  and  8,  the  values  of  £2  and  S±. 

In  like  manner  all  the  values  of  S2n  might  be  found  and  thence  the 
Bernoullian  numbers.  The  series  of  integrations  commenced  in  Arts.  332 
and  333  gives  another  method  of  calculating  S2n,  since 


10.  From  the  cosine-series    for  cosh  mx  (taking  l=n),  Art.  323, 
derive 


smh  rmt 
smh  mx— x 


2m2  sinh  ran    sin  x         sin  2X  sin 


^W2+I        2| 

and  show  that  this  result  agrees  with  the  value  of  sinh  mx  in  Art.  323. 
ii.  Trace  the  graph  of  the  quasi-periodic  function 

dx 

~ ; — o»         0<i. 

cos  x+az 

See  equation  (i),  Art.  312,  showing  that  one  of  the  points  farthest  from 
the  bisecting  line  y=x  is  on  the  line  y=K—  x.     Show  also  by  formula 

(G),  p.  41,  that 

(\ 
i  ~*  a  \ 
tan  *x], 
i  +  a             / 


and  discuss  the  case  in  which  a=i. 


§  XXI.  J  EXAMPLES.  367 

12.  Derive  the  expansion 

log  sin  $x=  —log  2—  (cos  x+%  cos  2x-\-^  cos  3^+  .  .  .); 

71 

and  thence  the  value  of       log  sin  \x  •  cos  nx  dx. 


f* 

log  sin  \ 

Jo 

f*  '  a  sin  x 

13.  Evaluate        tan"1  -  smnxdx. 

J0  i  —  a  cos  x 

<?          A     4 

bee  Art.  T.II. 

14.  Derive  the  expansion,  when  a<i, 

a  sin  #  cfo:  a2     a3 


2W 


tan 


/  a2  ^a»  \ 

—  (a  cos  x  —  g  cos  2;x;~l  —  2  cos  3^""  •  •  */' 


thence,  putting  0=1,  derive  the  result  of  Example  i,  and  show  why 
it  is  limited  to  values  between   —  TT  and  TT. 

15.  Show  that  for  all  values  of  x  between  ±'£TT  inclusive  of  these 
limiting  values 

4  /  .  i  i  \ 

x=  —  {  sin  x — 5  sin  T.X+—^  sin  zx—  ...  I, 
*  \  2T  52  I 

and  trace  the  complete  graph  of  the  second  member. 

1 6.  By  means  of  the  sine-series  for  x(rt  —  x)  show  that 

cos  33;    cos  ' 

~T~   ~? 

for  all  values  of  x  from  —  ^  to  £TT  inclusive  of  the  limits,  and  thence 
derive  the  numerical  series.  i       i       i  _ns 

I  o     I          o  o     1        "    •    •  « 

33     S3     73  32 

Compare  Diff.  Calc.  (Ex.  XX 111,  22).  This  is  also  the  result  of 
putting  x=%7i  in  Example  7,  and  the  others  of  the  same  set  result  from 
the  sine-series  for  higher  polynomials,  see  Example  8. 


368  DEFINITE  INTEGRALS.  [Ex.  XXI. 

17.  From  the  result  of  Example  4,  by  means  of  transformations 
and  the  trigonometric  formula 

•    (*      &\ 
cos  ^0— sm  ^0  =  <t/2  sin  ( ), 

\4      2  / 
show  that,  for  values  of  0  from  —  JTT  to  \TZ  inclusive, 

.             4 1/2 /sin  0     sin  30     sin  50  \ 

sin  £0  = (  -  -...); 

71       \    I.3  5.7  9-II  /' 

also  that  the  same  series  represents  cos  |0  from  6=^n  to  0=f~. 

18.  In  a  similar  manner  show  that 

K    \2        3-5         7-9  "  / 

when  0  is  between  —  \TI  and  \x,  and  that  this  series  always  represents 
the  greater  of  the  two  quantities  sin  \Q  and  cos  \Q  taken  positively. 
The  equations  derived  in  like  manner  from  Example  5,  or  by  taking 
derivatives  of  those  above  are,  for  values  between  ±£TT, 

.  „      84/2  /2  sin  20     4  sin  40  ,  6  sin  60  \ 

sm#/=  —  .  .  .  1, 

^    V    3-5  7-9  11-13  / 

ia_  84/2  /cos0     3  cos  30  ,  5  cos  50  \ 

COS  jC/  —  I  ...    I . 

71       \    1-3  5.7  9-II  / 

in  the  first  of  which  the  limits  are  excluded. 

19.  Expand   cos  mx  in  cosines  of  multiples  of  x,  and  verify  the 
result  in  the  cases  where  m  becomes  o  or  an  integer. 

sin 
cos  mx=- 


r                 „  /    COS  X         COS  2X        COS  T.X  V~| 

I  +  2m-i(-^ 5 2 5+    o      '     2~*  •  •    /     I' 

\i2— w2     22—m2    3*— fir  /J 


W7T 

20.  Expand  sin  w^c  in  a  cosine-series  for  values  of  x  between  o  and  ~. 
i  — cos  wwf"  „/  cos  2X     cos  4^      cos  6x 


—  zm- 


I                   0/COS2^        COS  AX        COS  6X  \'l 

I  —  2m^(-^ 2  +  -^ 2+22 2+    •   •   •  ) 

L  \a*— m*    4^— w^    fr—nf*          /J 

/  COS  ^C  COS  3*         COS  5#  \ 

I       o  o    I     ~o  o    I        o  o    1       •    •    •      I  * 

\i  —  w^     3J  —  mz     y—mt  / 


sin  mx — 

+  cos  mrr  I  cos  x    .   cos  T.X  .  cos 


§  XXL]  EXAMPLES.  369 

When  m  is  an  integer,  one  line  (inclusive  of  the  term  which  takes 
the  indeterminate  form)  vanishes. 

21.  Expand  cos  mx  and  sin  mx  in  sine-series. 

1  4-  cos  nm  I  sin  x       3  sin  T.X     <  sin  =53: 


cosmx=2 


3  — 


i  —cos  WTT  /2  sin  2X    4  sin  ^x  \ 

'   ^ '.  \  ~2  2~          2  2    '     •  •  •    ) 

71         \2  —m        4  —  m  j 

2  sin  m,7c  I  sin  x       2  sin  2#     T.  sin  ?#  \ 

sm  w#=  — ~ \~2~2 — 2 2_    2  +^ — "V"  .  .  .  ). 

TT       \i      w       2      m       3  — m  / 

22.  Find  the  sine-series  for  cos  3^,  and  thence  show  that  from 

3        8  /     sin  x     2  sin  2X     T.  sin  ix  \ 

cos^#=-( H —        -+- ^-+...1. 

2       x\       5  i-7  3-9  / 

23.  If  for  values  between  o  and  TC  inclusive 


n  cos  nx    and    ^(^)  =  C/+^I^4«  cos 

«=  i  n=  i 

prove  that 


24.  Show  by  means  of  Example  23  and   equation  (5),  Art.  311, 
that,  when 


I"*  /        a4     a6  \ 

[log  (i  —  2a  cos^+a2)]2J^=2^(a2H — ^-\ — ^+  . . .  ). 

Jo  V  22      32  / 

Prove  that 

f*  dx  _     i  +  a2 

J0(i  — 2acos^+a2)2       (i  — a2)3' 


26.  Prove  that 

i^7?    F7?     S27? 


370  DEFINITE  INTEGRALS.  [Ex.  XXL 

27.  Prove  that 

JT  JT 

[ 2  (log  cos  9)2  d6=  f  *  (log  sin  <W  dd>=n(l°g  2^  +  - 

Jo  Jo  2  24 

(see  Art.  321  and  Ex.  12);  also  that 

JT 

fT  ,    ,  .       ,     ,  ,       7T(log  2)2       7T3 

log  cos  9 •  log  sin  d)  da>=  . 

Jo  2  48 


•-/T.2 


28.  Putting  <b(x)  =  -  —  ~  in  the  Theorem  of  Example  2-1, 

i  —  2acosx+a? 

prove  that,  when  a<i, 


and  thence  that,  when  a  approaches  unity,  the  limiting  value  of  the  first 
member  is  TT/~(O). 


log  (l  —  26  COS  #+  fr2)  ,    _  27T  log  (l  —  O 


29.  Prove  that 

plogji 

J0     i  —  2acos#+a2  i  — 

30.  If  for  values  of  x  between  o  and  TT 

«=«  M=« 

f(x)  =  2  Bn  sin  nx    and     <p(x)=  2  B'n  sin  nx, 
prove  that 

fV  --Mi°°      ' 

J°  2n=i 

31.  Prove  that 


by  direct  integration  and  also  as  a  case  of  the  Theorem  of  Example  30. 


§  XXL]  EXAMPLES.  371 

f  *          sin2  x  dx 

T>2,  Evaluate       -. —          ; — ^>,  when  a<i,  and  when  a>i. 

Jo  (1  +  20,  cos  x+a2)2 


2(1 -a2)'     2a2(a2-i)' 

33.  By  means  of  Examples  23  and  30  and  the  trigonometric  for- 
mula cos  2mx=co&2  mx—  sin2  mx,  derive  the  result 

7T  COt  WlTl          I  I  I  I 


2m         2m2     i2  —  m2     22—m2      2  — 


34.  Prove  in  like  manner 


TT  coth  mn  _    i  i  i 

2w          2m2 


35.  Show  that 

f  **  9  i  j          ^  l°g  2     n  <? 

x2  log  cos  x  dx=  --  ~  ---  $3, 
Jo  24          4 

f**    2  I  '  J  7T3log  2       37T 

|    x2  log  sin  #  ^=  --  ^—  +  ^^3, 

f  *    2  1       •       j          ^r3  log  2     ^ 
x2  log  sin  #  dx=  -----  03, 

Jo  32 

where  53  =  iH  —  oH  —  5+~5+  •  •  • 
26    3^    4d 

36.  Construct  a  cosine-series  whose  value  shall  be  +1  from  x=o 
to  x=a,  zero  from  x=a  to  ^=^—  a,  and—  i  from  x—n—  a  to  «;=7r. 
Verify  the  statement  in  Art.  338  with  regard  to  the  value  of  /(a). 

f(x)  =  —  [sin  a  cos  x+—  sin  30:  cos  $x-\  —  sin  50:  cos  5#+  .  .  .  ]. 

n  3  5 

37.  Show  that  the  integral  of  the  series  in  Example  36,  namely 
—  (sin  a  sin  x-\  —  ^  sm  3a  sm  3^+"^  sin  5a  sm  5^+  •  •  •)> 

represents  x  from  x=o  to  ac=o:,  a  from  x=a  to  ^=TT—  a,  and  ^—  x 
from  #=TT—  o;  to  x=n. 


372  DEFINITE  INTEGRALS.  [Art.  341. 

XXII. 

The  Eulerian  Integrals. 

341.  A  definite  integral  with  constant  limits,  while  not  a  func- 
tion of  the  variable  whose  differential  appears  under  the  integral 
sign,  is  in  general  a  function  of  any  other  algebraic  quantity 
which  occurs  in  its  expression;    and,  if  that  quantity  admits  of 
continuous  variation,  the  integral  is  a  continuous  function  of  it, 
and  admits  of  differentiation  with  respect  to  it.     For  example, 

in  the  integral      xndx,  n  admits  of  any  value  greater  than  —  i ; 

J  o 

therefore  the  integral  is,  for  the  range  of  values  from  —  i  to  +00, 
a  continuous  function  of  n.  In  this  case,  the  value  of  the  inte- 
gral as  a  function  of  n  and  of  its  derivatives  with  respect  to  n 
are  readily  expressed  by  means  of  the  elementary  functions.  See 
Art.  278. 

Certain  ones  among  those  definite  integrals  which  are  not  thus 
"integrable"  in  elementary  functions  have  come  to  be  regarded 
as  fundamental  ones,  and  serve  to  define  new  functions  which 
are  employed  in  the  expression  and  calculation  of  other  integrals 
of  more  complex  form.  Of  these  the  most  important  are  those 
known  as  the  First  and  Second  Eulerian  Integrals. 

342.  The  first  Eulerian  Integral  is  a  function  of  two  variables 
denoted  by  B(m,  n),  and  hence  sometimes  called  the  Beta  Func- 
tion.   It  is  defined  by 


B(mt  w)=     xm- 

J  o 


in  which  each  of  the  exponents  must  be  greater  than  —  i ;  there- 
fore m  and  n  are  restricted  to  positive  values.  By  Art.  97, 
B(n)m)=B(m,n),  so  that  m  and  n  are  interchangeable.  The 


§  XXIL]  THE  EULERIAN   INTEGRALS.  373 

value  of  the  integral,  when  one  or  both  of  the  variables,  m 
and  n,  is  an  integer,  is  expressible  in  elementary  functions; 
it  may  be  found  by  a  formula  of  reduction,  or  directly  as 
follows : 

Let  n  have  any  positive  value,  then 

f'  i  fJ  i 

xm~1dx=—        and  xmdx= — ; — . 

J0  m  J0  m  +  i 

Subtracting, 

f1  ITT 

\   Xm-I(l-X}dx  =  — =  — ; — :.        .      .      .      (i) 

J0  m    m  +  i     m(m  +  i) 

In  like  manner,  putting  m  + 1  in  place  of  m, 


and  subtracting  this  from  equation  (i), 

f  xm~*(-L  —X}2  = - —  —  — = 

J0  ;      m  +  i[_m    m  +  2j 

Again,  putting  m  +  i  in  place  of  m, 


and  subtracting, 

t 

Xm~l(l—x)3dX  =  -f r-; -J 


2-3  /    x 

•     ...       (3) 


374  DEFINITE   INTEGRALS.  [Art.  342. 

It  is  evident  that,  in  this  manner,  we  can  prove 
f1  n\ 


. 
m(m  +  i)  .  .  .  (m  +  ri) 


...     (4) 

for  all  integral  values  of  n.     This  equation  gives  the  value  of 
B(m,  w  +  i)  when  n  is  an -integer. 

Gauss'  JI  Function. 

34-3.  When  m  is  fixed  and  n  increases  without  limit,  B(m,  n) 
approaches  zero  as  its  limit;  but  its  ratio  to  n~m  is  found  to  be 
finite.  This  ratio  constitutes  a  function  of  the  single  variable 
m  known  as  the  Second  Eulerian  Integral. 

The  limiting  ratio  in  question  may  evidently  be  derived  from 
equation  (4)  above  by  making  the  integer  n  increase  without 
limit.  First  transforming  the  integral  by  putting  x  =  y/n,  the 
equation  becomes 


i  P         /       y\n         i       i 

ym~l   i--)  dy  = — 

nm]  0          V       n  /  m  i  +m 


+m         n  +  m 

"When  n  is  made  infinite,  we  have,  by  evaluation  of  an  indeter- 
minate form, 


Ml    -<-'• 

\       nl  JM=oo 


Hence  the  equation  may  be  written  in  the  form 

_  T  r     *     2        w 

•y     mL 


I-.'  '  "> 


Here  the  first  member  has  a  finite  value,  while  in  the  second 
member  the  factor  nm  becomes  infinite,  and  the  continued  prod- 
uct (in  which  each  factor  is  a  proper  fraction)  has  zero  for  its 


§  XXII.]  GAUSS'   n  FUNCTION.  375 

limit  when  the  number  of  factors  is  infinite.     The  integral  (in 
which  m  must  be  positive)  is  the  Second  Eulerian  Integral. 

344.  Gauss  denoted  by  the  symbol  77  the  function  denned  by 


-.--J— •  •  '  (I) 

Thus  the  equation  above  may  be  written 

f  ^  /  /  /       \ 

ym-ie-y(ly= /2\ 

J0  m 

He  also  defined  II(  —  m)  as  the  result  of  changing  the  sign  of  m 
in  equation  (i),*  so  that 

P  123  n    n 

H(-*w)H»-*~ — - — f 3-       .    .   (3) 

I—  •*•  o  _l  n  =  oo 

In  this  last  expression,  the  factor  n~m  vanishes,  and  the  con- 
tinued product  becomes  infinite,  when  n  is  infinite. 
Now  multiplying  equations  (i)  and  (3),  we  find 


f     i          22          T?  n2     "It 

-m)  = 31 22        2-  •  •  ~     — 

LI  —  m*  2*  —  m2  y  —  m2        r^  —  ?#2JM=,  OT 


(4) 


*  It  is  to  be  noticed,  however,  that  the  integral  in  equation  (2)  does  not  admit 
of  negative  values  of  m. 

f  The  fractions  in  equations  (r)  and  (3),  taken  together,  form  a  series  running 
to  infinity  in  both  directions.  In  equation  (4),  an  equal  number  of  factors  is 
taken  from  each,  and  then  the  number  is  made  infinite,  the  object  being  to  get 
rid  of  the  power  of  this  number  before  it  is  made  infinite.  This  could,  however, 
be  done  by  taking  n  factors  from  the  series  (3)  and  rn  factors  from  the  series  (i), 
r  being  constant;  for  (rri)mXn~m  reduces  to  r* ,  and  thus  remains  finite  when 
n  is  made  infinite.  Thus  the  product  II(m)II(-jM)  may  be  written  in  the  form 

mT     n         n—  i                 i          i         2                rn    ~\ 
r    \ • .  .  . .  .  . 

\_n  —  m  n—  i  —  m         i  —  mi  +  m2  +  m         r;H-wJn=oo 


This  expression  has  therefore  a  value  independent  of  r.  Supposing  r>i,  the 
expression  contains  an  infinite  number  of  extra  factors  on  the  right.  It  follows 
that  the  product  of  these  factors  must  be  r~m.  See  the  note  on  p.  234  Diff. 
Calc.,  in  which  the  series  is  the  reciprocal  of  that  here  considered. 


376  DEFINITE  INTEGRALS.  [Art.  344. 

Putting  W  =  <£/TT,  the  reciprocal  of  the  second  member   becomes 
the  continued  product 


where  the  number  of  factors  is  infinite.     The  value  of  this  is, 
by  DifL  Calc.,  Art.  234,  sin  <£/<£.     Therefore 


. 

sm  <f>     sin  mn 


This  equation  expresses  a  fundamental  property  of  the  function 

II,  and  shows  that  H(—m)  is  finite  except  when  m  is  an  integer. 

345.  If,  in  equation  (i),  we  put  m  +  i  in  place  of  m,  we  have 


- 


hence,  comparing  with  equation  (i  ),  we  have 


),     .     .     (6) 


a  second  fundamental  property  of  the  function. 

Putting  w  =  o  in  equation  (i),  we  have  U(o)  =  i;  whence,  by 
equation  (6),  we  derive  successively,  77(i)  =  i,  77(2)  ==2!,  and  in 
general  when  p  is  a  positive  integer  TL(p)=p\.  Thus  the  func- 
tion is  a  generalization  of  the  factorial  product;  this  is  in  fact 
directly  evident  on  giving  to  m  in  equation  (i)  an  integral  value. 


§  XXII.]  THE  GAMMA    FUNCTION.  377 


The  Gamma  Function. 

346.  The  definite  integral  in  equation  (2),  Art.  344,  is  the 
Second  Eulerian  Integral,  and  according  to  Legendre's  notation 
is  denoted  by  r(m)\  it  is  therefore  generally  called  the  Gamma 
Function.  Thus 


,0 

r(m)=\ 

J 


*e-xdx\  ......     (i) 

and,  comparing  with  Gauss'  notation, 


(2) 


Suostituting  m  equation  (6)  of  the  preceding  article,  we  have  the 
fundamental  relation 


(3) 


which  is,  in  fact,  equivalent  to  the  formula  of  reduction  given  in 
Art.  92.     We  have  therefore  also 


(4) 


so  that  a  table  of  the  values  of  the  T-function  is  also  a  table  of 
values  of  the  ./I-function. 

Equation  (2),  together  with  the  general  definition  of  II  (m), 
serves  to  define  F(m)  for  negative  as  well  as  positive  values  of  m. 
Equation  (3)  may  also  be  written  in  the  form 


n  — 


Again,  by  equations  (3)  and  (4),  the  fundamental  property 


DEFINITE  INTEGRALS. 


[Art.  347. 


expressed  in  equation  (5),  Art.  344,  becomes 

71 


sin 


(6) 


tions  (3)  and  (5), 
—  I)  =  ffX    etc. 


347.  Equation  (i)  gives  at  once  r(i)  =  i;  whence,  by  suc- 
cessive applications  of  equation  (3),  F(p)  =  (p  —  i}\,  when  p  is  a 
positive  integer.  By  equation  (5),  r(o)=oo,  and  F(ri)  is  also 
infinite  for  all  negative  integral  values  of  n.  Putting  m  =  \  in 
equation  (6),  we  derive  F  ^)=  |/TT;  whence,  again  using  equa- 
(f) =  ii*r,  r(|)  =  |^,  etc.,  T(-|)=  -2^, 
These  special  values  serve  to  show  that 
the  graph  of  the  function  y  =  F(x] 
has  the  general  form  given  in 
Fig.  63,  the  number  of  branches 
for  negative  values  of  x  being 
infinite. 

By  equations  (3)  and  (5),  the 
/"-function  of  any  number  is 
readily  expressed  in  terms  of  a 
value  of  F(n),  in  which  n  is 
between  i  and  2;  it  is  therefore 
only  necessary  to  tabulate  F(ri) 
for  this  interval,  corresponding 
to  the  arc  AB  in  the  diagram. 

For  purposes  of  computation, 
a  table  of  values  of  logic  r(n)  is 
more  useful.  Such  a  table,  car- 
ried to  12  decimal  places,  was 
constructed  by  Legendre.*  An  abridgement  of  this  table  to  5 
decimal  places  is  given  at  the  end  of  this  chapter. 


y= 


FIG.  63. 


*  Traite  des  Fonctions  Elliptiques  et  des  Integrates  Euleriennes,  vol.  ii,  pp 
490-499. 


§  XXII.]          FORMS   OF    THE   EULERIAN    INTEGRALS. 


379 


Transformations  of  the  Eulerian  Integrals. 

34-8.  A  variety  of  definite  integral  forms  of  the  Gamma  func- 
tion results  from  changes  in  the  current  variable,  with  correspond- 
ing changes  in  the  limits.  For  example,  putting  z  =  e~x,  whence 
x  =  —log  z,  we  find 

f°°  f  I"/       I\K-' 

/"»=     xn-le-*dx=\  (-logz)M-'dz  =      (log-)       dz*     (i) 

Jo  J  o  J  o  \  Z  / 

Again,  putting  x=z2,  we  have 

f°° 
T(w)  =  2     e-*z2n~ldz  .......     (2) 

^  o 

This  form  gives  a  direct  evaluation  of  r(|);  for,  by  Art.  288, 

f°° 
=  2     e-^0=  fa 

J 


agreeing  with  Art.  347. 

Putting  x=ay,  where  a  is  any  positive  quantity,  we  find 

r(ri)=an\   yn-Ie-°ydy  ......     (3) 

•*  o 

349.  If,  in  the  first  Eulerian  integral,  we  put  x  =  i/z,  we  have 
B(mtn)=[*»-<(i-xr-'dx  =  l    (S~?*n  '  dz.  .     .     (i) 

J  o  J  I  Z 

Again,  putting  z=y  +  i  in  the  last  member, 


*  The  last  member  is  the  original  form  in  which  Euler  and  Legendre  treated 
the  Gamma  function. 

f  It  will  be  found  that  substituting  the  reciprocal  in  this  form  merely  inter- 
changes m  and  n.  The  three  forms  in  equations  (i)  and  (2)  are  the  simplest 
forms  of  B(m,  n).  See  also  Ex.  24  below. 


380  DEFINITE  INTEGRALS,  [Art.  349. 

Putting  x/a  for  the  current  variable  in  each  case,  these  three 
forms  are  rendered  homogeneous  in  x  and  a;   the  results  are 


,n'),    ....     (3) 
B(m,  n) 


A  useful  transformation  results  from  putting  x  =  sin2  6,  whence 
i  —  #  =  cos2  6  and  dx  =  2  sin  6  cos  0  <#?.     We  thus  find 


B(m,  n}=2itfm-1  6  cos^-^ddd,      ...     (5) 

J  o 

in  which  m  and  n  may  have  any  positive  values. 

350.  In  the-special  case  where  m+n  =  i/  the  5-function  in 
the  form  (2)  has  already  been  integrated;  thus,  by  equation  (2), 
Art.  302, 

B(m,i-m)  =  -  --  ,   .     .    .    «    .         (i) 

sin  mn 

in  which  o<w<i.     Using  the  integral  forms  in  equations  (i) 
and  (5)  above,  we  have  therefore  also 

r      dx         r    dz        n 

J0xI-fn(i-x)m~Jlz(z-i)m~smmK     *    *    '    *** 
and 

X 

r*  n 

t&K2m-l6dd  =  —  :  -  ........     (3) 

J  0  2  sin  nm. 

As  a  particular  case,  putting  m  =  J  or  f  in  equation  (3), 


f?      ^          f 

^T  —  7;= 
J0   i/tan/9     J0 


-—  . 
4/2 


§  XXII.]  RELATION  BETWEEN  THE  EULERIAN  INTEGRALS.    381 


Relation  between  the  two  Euierian  Integrals. 

351.  The  product  of  two  /^functions,  when  put  in  the  double 
integral  form,  gives  rise  to  a  relation  between  the  two  Euierian 
Integrals.  We  shall  derive  this  relation  by  two  methods  analogous 
to  those  employed  in  the  evaluation  of  k2  in  Arts.  289  and  288 
respectively. 

Taking  a  as  the  current  variable,  we  have 


,00 

r(m)=\    am~re-ada. 

J  o 


Multiplying  each   element  by  the  constant   F(n)  in  the  form, 
see  Art.  348, 


r(ri)=an\   ofn-te-axdxt 

J  a 


we  have 

ff.  »00 

<cn~*e~axdxda. 


Reversing  the  order  of  integration,  we  have 

f  00    /*00 

r(m)r(ri)=\        am+n-le-^x+l) 

*  o  •  o 

Performing  the  a-integration  by  equation  (3),  Art.  348, 

r  r(m  +  n]  J30    xn~*dx 

nm)r(»)-]o(-^^^^ 

Hence,  by  equation  (2),  Art.  349, 

..•:...  w 


382  DEFINITE   INTEGRALS.  [Art.  352. 

352.  In  the  other  method  of  deriving  this  relation  we  take 
two  independent  variables  x  and  y,  and  use  the  form  (2),  Art. 
348.  Thus 

0-     « 
c-*-*x*m~*ya* 
o 

Then  putting  x  =  r  cos  6  and  y  =  r  sin  6, 


=  2  1  2  r(m+n)  cos2W-'0  sin2"-  '0  ^. 

J  o 

Hence,  by  equation  (5),  Art.  349, 

B(m,  n}  = 

x 


as  before.     When  w+t  =  i,  this  equation  gives,  by  means  of 
equation  (6),  Art.  346,  another  proof  of  equation  (i)  of  Art.  350. 

Reduction  of  Integrals  to  Gamma  Functions. 

353.  By  means  of  the  equation  proved  above,  equation  (5), 
Art.  349,  may  be  written 


f' 

J0 


'tf  sin 


putting  in  this  2in  —  i=p,  2n  —  i  =q,  it  becomes 


fa  \     2     /       \     2     , 

cos  0  sin  0  dd  =  -        /,....  .\ — » 

Jo 


§  XXIL]     REDUCTION  OF  INTEGRALS  TO  F-FUNCTIONS.     383 

in  which,  since  m  and  n  must  be  positive,  p  and  q  must  each  ex- 
ceed —  i.  This  is  a  generalization  of  formula  (Q),  p.  126,  with 
which  it  can  be  shown  to  agree  in  the  several  cases  where  p  and 
q  are  integers. 

In  its  application,  the  arguments  of  the  /""-functions  are  to  be 

reduced  by  the  formula  7»  =  (r-i)/>-i)  or  7»=-/>  +  i) 

until  they  fall  within  the  limits  of  the  table.     For  example,  to 
compute  the  integral  when  p  =  %  and  <?  =  §,  we  have 
i(?+i)=f     Then 


•-j; 


2/W)       a.|r(i) 

_i8r(i.333...)r(i.833...) 

35r(i.i66 . . .) 

Using  the  table  on  p.  401,  the  computation  takes  the  form 

log  18  =  1.25527 
log  r(i. 333^=9.95084-10 
log  ^(1.833!) -9.97343- 10 

colog  35  =8.45593-10 
colog  77(i.i66§)  =0.03258 

log  7  =  9.66805-10        .*.  7=0.46564. 

Besides  the  case  in  which  p  and  q  are  both  integers,  there  are 
two  others  in  which  the  result  is  free  from  /^-functions.  These 
are,  first  when  p  or  q  is  an  odd  integer,  in  which  case  one  of  the 
T's  is  a  factorial  and  the  others  after  reduction  divide  out.  (Com- 
pare Ex.  VII,  1 8.)  Secondly  when  p  +  q  is  an  even  integer,  in 
which  case  advantage  may  be  taken  of  equation  (6),  Art.  346. 
For  example, 


„.  ,.  - 

2T(2)  12  12  Sin  $7t       6 


384  DEFINITE  INTEGRALS.  [Art.  354. 

f  °°  cos  bz  ,  f  °°  sin  bz 

354.  The    integrals         —  -  —  dz    and          —  ^—  dz,    where 

Jo         Z  Jo        2 

o<w<i,  can  be  evaluated  by  forming,  as  in  Art.  351,  a  double 
integral,  and  then  reversing  the  order  of  integration.  Thus,  putting 
U  for  the  first  of  these  integrals,  and  multiplying  its  elements  by 
F(n)  in  the  form 

,00 

F(ri)=zn      e-xexn~Idxt 

J  o         • 

equation  (3),  Art.  348,  we  have 

=0  ,00 

F(n)U=\    cos  for     e-*zxn-'ldxdz. 

Jo  J  o 

Reversing  the  order,  the  z-  integration  can  be  performed  by  Art. 
281,  and  then  the  *-  integration  by  equation  (4),  Art.  302.  Thus 

r      r  r  x»dx 

F(n)U  =     xn~*\   e-*z  cos  bz  dzdx=\   -^—  ^, 

J0  Jo  J  o  X  -ru" 

and  putting  x  =  by, 

rcosbz        b«~*  t*y«dy  _        Kb"~l 

J0    zn    dz~r(n))0y2  +  i~2r(n)cos%n7c'      * 

f^sin  bz  , 
In  like  manner,  putting  V=       ~^~dz  and  supposing  b  positive, 

Jo       Z 

r     r  r  bxn-*dx 

)V*=     xn~l     e~xz  sin  bz  dz  dx=          2    ^ 

J  0  J  o  J  0     X  -rD* 

[*yn~*dy        Tzbn~l 

_Jn-i       Z  --  1.  —  -  , 

J  0   y2  +  1      2  sin  \mi 

r°°sin  bz  nb"-* 

J0    zn  2F(ri)  sin  \mi 


we  find 


§  XXIL]     REDUCTION  OF  INTEGRALS  TO   F-FUNCTIONS.     385 

If  b  is  negative,  this  integral  changes  sign.     In  the  limiting  case 
when  «  =  i,  the  result  agrees  with  Art.  283. 

355.  Two  integrals  of  a  more  general  form  result  from  the 
substitution  of  a  complex  value  for  the  constant  a  in  equation 
(3),  Art.  348.  Thus,  putting  a  =  a-ib  in 

r 

r(m)=am\    e-axxm-'dx, 

J  o 

we  have,  since  e^^cos  bx+i  sin  bx, 

r00  r(m)     r( 

e~ax(cos  bx+i  sin  bx)xm-1dx  =  f  —  -T~  ^^ 
Jo  (a—  ID)          p 

where  a+ib=p(cos  6+i  sin  6},  so  that 

an(i     0  =  tan-  '6 


% 

Expanding  by  DeMoivre's  Theorem,  and  separating  real  and 
imaginary  parts, 

r  r(m)  r          -b~] 

e~ax  cos  bx  .  xtn~Idx=  -•  —    ^^  cos    m  tan   J-   ,  .     (i) 

Jo 


and 


e-axsmbx.xm~Idx  =  -  —*-m  sin  \m  tan-I-    .    .     (2) 

•  (««+«•)"  aj 

In  these  equations,  a  is  positive  and  m>o.     They  include  the 
general  equations  given  in  Exs.  XIX.  6  and  8. 
356,  When  a  =  o,  equations  (i)  and  (2)  become 


386  DEFINITE  INTEGRALS.  [Art.  356. 

and 

f00  F(m\   .       7t 

xm~ '  sin  ox  dx  =  —r-—  sin  m-t 
Jo  *>m  2 


but  in  this  case  we  must  have  w<  i. 

Putting  m+n  =  i,  these  may  be  written 


and 


sn  » 

2 


cos 

•* 


in  which,  since  m  cannot  be  negative,  n  cannot  exceed  unity. 
Since  T(i  —n}=  r(  . 

\      /    ' 

with  those  of  Art.  354. 


Since  F(i—n)=  „,  . — : ,  these  last  equations    are  identical 

'     i  (n)  sm  mi 


Reduction  of  Certain   Multiple  Integrals. 

357.  The  double  integral  of  x1~lym~^dy  dx,  where  x  and  y 
have  all  positive  values  such  that  x  +  y<a  is  expressible  in 
/""-functions.  Thus 

J+'BQ, 


fa   ra-x  j    fa 

x1-  Iy»t-  *dy  dx  =  — 

J  o  J  o  m^  c 

by  equation  (3),  Art.  349.     Hence,  by  equation  (i),  Art.  351, 

p  r~* ;-, 

^o  Jo 


§  XXII.]  REDUCTION  OF  CERTAIN  MULTIPLE  INTEGRALS.  387 

This  result  may  be  extended  to  any  number  of  variables. 
Thus,  for  three  variables,  putting 


U 


fa    ra  —  Xfd  —  x  —  y 

=\  xl-*ym-*zn-l 

J  o  J  o        Jo 


the  y-  and  2-integrations  can  be  performed  by  equation  (i),  a—  x 
taking  the  place  of  a;  hence 


)  fa 
\\   x      (a — x}m+ndx 

t  )J  o 


m+n 


by  equation  (3),  Art.  349.  Therefore,  if  x,  y  and  z  are  three 
variables  subject  to  the  condition  that  each  is  positive  and  their 
sum  shall  not  exceed  a, 


,  .  („ 


It  is  obvious  that  this  result  can,  in  like  manner,  be  extended 
to  any  number  of  variables  subject  to  the  same  conditions.     Thus 

U=\    \  .  .  .^-y-'z"-1  .  ..  dzdydx     ___ 


.„... 

358.  By  means  of  equation  (3),  an  integral  of  the  form 

.).  .  -dzdydx 
can  at  once  be  reduced  to  a  simple  integral. 


388  DEFINITE  INTEGRALS.  [Art.  358. 

For  this  purpose,  we  establish  the  following  general  theorem: 
Let  oU  denote  the  element  of  a  multiple  integral  U,  and  let  V 
be  the  integral  under  the  same  limits  of  the  element  f(a)3U, 
where  a  is  a  function  of  the  variables  of  integration:  then  if  U 
is  expressible  as  a  function  of  a,  we  shall  have,  with  proper  limits 
for  a, 

V=\U'f(oL)da,    where     U'  =—-. 

To  prove  this,  imagine  the  multiple  integrals  U  and  V  to  be 
transformed  to  any  new  set  of  variables  of  which  a  is  one,  and 
suppose  all  the  integrations  except  that  for  a  to  have  been  per- 
formed. The  expression  under  the  final  integral  sign  will,  by 
hypothesis,  be  U'da  in  the  first  case.  Now,  since  in  these  inte- 
grations a  is  treated  as  a  constant,  the  integrations  in  the  second 
case  will  not  be  affected  by  the  presence  of  the  factor  f(a}  ;  hence 
the  expression  under  the  final  integral  sign  will  be  Ur  f(a}da\ 

therefore  V  =    U'f(a)da,  as  stated  above.* 

In  the  present  case,  U  is,  by  equation  (3),  a  function  of  a, 
and  putting  for  abridgment 


)  •••  dU 

AT  —      v  '      v      '  ---  114.^4.  .  .  ,\  fJn 

~(H 


.hence,  by  the  theorem 


*  In  further  explanation  of  the  principle  employed  above,  see  Art.  222,  where 
it  is  applied  to  a  problem  of  mean  values.  In  that  article  A/",  "the  number  of 
cases,"  corresponds  to  U,  and  MN  "the  aggregate"  to  V;  TV  is  a  known  function 
of  r  (which  corresponds  to  a),  and  M0  (also  a  function  of  r)  corresponds  to  /"(a). 
Then,  when  the  independent  variable  r  is  increased  by  dr,  the  number  of  "new 
cases"  is  dN  and  the  new  part  of  the  aggregate  is  M0dN;  whence  MAT"  is  ob- 
tained by  integration  of  M0dN  with  respect  to  r. 


§  XXII.] 


THE  FUNCTION    LOG 


389 


The  Function  LogT(\  +x),  and  Eulers  Constant. 
359.  By  equation  (i),  Art.  344, 


I+XI 


i 
x 


whence 


log  F(i  +x)  =  I  x  log  n — log  (i  +x)  —  log  (i  +  %x)  —  •  •  • 


Developing  the  logarithms  by  the  series 

log  (i  +  x)  =  x  — 

and  collecting  like  terms,  we  find 


.  (i) 


log 


log 

l_ 


The  logarithmic  series  employed  are  convergent  when  x<i, 
and  the  numerical  series  are  all  convergent  with  the  exception  of 
that  which  occurs  in  the  coefficient  of  x,  which  is  known  as  the 
harmonic  series  and  is  shown  in  Diff.  Calc.,  Art.  180,  to  have  an 
infinite  value.  Thus  the  coefficient  of  x,  which  takes  the  form 


390  DEFINITE  INTEGRALS.  [Art.  359. 

oc  —  oo  ,  must  have  a  finite  value.  The  numerical  value  of  this 
coefficient  (which  is  found  to  be  negative)  is  known  as  Eukr's 
Constant  and  is  generally  denoted  by  f.  Thus 


r    i    i 

I+2~+7 

l_         2 


its  value  will  be  found  in  Art.  362. 
360.  Putting,  when  n>i, 


i       i 

+  "+    *~*~  '  ' 


the  development  of  log  F(i  +  x)  may  now  be  written 

"      .    .     (i) 


Again,  since  the  sign  of  x  may  be  changed  in  the  expression  for 
n(x),  Art.  344,  we  have  in  like  manner 


log  r(i  -oc)  =  r*  +  %SzX2  +  %S*x?  +  ••-     .    .     .     (2) 

The  sum  of  the  terms  containing  even  powers  of  x  in  these 
series  can  be  at  once  evaluated.  For,  by  equations  (3)  and  (6), 
Art.  346, 


sm 
therefore  the  sum  of  equations  (i)  and  (2)  gives 


.     .     (3) 


*  Compare  Diff.   Calc.,   Art.   230-     In  Art.   244  the  even-numbered  S's  are 
shown  to  be  connected  with  Bernoulli's  numbers. 

f  This  equation  is  identical  with  equation  (2),  Diff.  Calc.,  Art.  239,  of  which 


§  XXII.]  EULER'S  CONSTANT.  391 

Substituting  in  equation  (i), 


7COC 


log  r(i  +*)=*  log  sl—  --(^+^3^+15^+  •  •  •),     (4) 

in  v.hich  the  sign  of  x  may  be  changed,  and  the  limits  of  con- 
vergence are  ±i. 

361.  The  values  of  Sn  all  exceed  the  first  term,  which  is 
unity;  hence  the  series  (4)  converges  slowly.  But,  if  we  put 
Sn  =  i  +  sn,  the  part  corresponding  to  the  first  term  takes  a  known 
form.  Thus,  substituting  in  equation  (4),  we  have 


in  which  the  sum  of  the  first  series  is,  by  DifL  Calc.,  Art.  198, 

i  i      I+x 

*  log  --  x.    Hence 

&  i—x 

7C3c(  'T  ~~"  $c  ] 

log  r(i  +X)  =  \  log  —  -          -  +  (i  -  f)x 

&  (i  +x)  sin  TIX     v 

-[K^  +  K^+.-.l     (5) 

and 


log  r(i  -*)-  j  kg 

••]•*    (6) 


the  differential  [equation  (2),  Art.  243]  gives  the  values  of  S2n  in  terms  of  Ber- 
noulli's numbers. 

*  By  means  of  these  series  Legendre  (having  first  calculated  the  values  of  Sn 
to  1  6  decimal  places,  together  with  the  value  of  f)  calculated  the  values  of  log  F(a). 
Since  it  was  only  necessary  to  cover  a  range  of  unity  in  the  values  of  the  argu- 
ment, only  values  of  *  less  than  |  had  to  be  used,  and  in  fact  other  relations  were 
used  to  limit  the  range  still  further.  See,  for  example,  Ex.  4  below. 


392  DEFINITE  INTEGRALS.  [Art.  362. 

362.  When  known  values  of  the  F-i  unction  are  substituted  in 
these  equations,  relations  between  7-  and  the  values  of  sn  are 
found.  Thus,  putting  x  =  i  in  equation  (5),  we  have,  by  evalua- 
tion of  an  indeterminate  form, 

r  =  i-ilog2-|$s3  +  to  +  }*7+"-]    ....     (7) 
Again,  by  putting  x  =  \  in  equation  (6),  we  derive,  since 


(8) 


l 

3-22       5-24       7.2 


A  table  of  the  values  of  sn  is  given  at  the  end  of  this  chapter, 
by  means  of  which  the  value  of  7-  *  to  10  places  of  decimals  will 
be  found  to  be 

^•=0.5772156649. 


The  Logarithmic  Derivative  ofr(jx). 

363.  Let  r'(x)  denote  the  derivative  of  r(x),  and  ^>(#)  the 
logarithmic  derivative;  then,  differentiating  equation  (i),  Art. 
359  (and  putting  x  in  place  of  i  +#),  we  have 


*  The  value  of  7-  has  been  calculated  by  Professor  J.  C.  Adams  to  263  places 
of  decimals,  Proceedings  of  the  Royal  Society  oj  London,  vol.  xxvii,  p.  94.  The 
method  involves  the  direct  summation  of  a  large  number  of  terms  of  the  har- 
monic series  and  calculation  of  the  remainder  by  means  of  Euler's  formula  for 
the  summation  of  series.  The  same  method  was  used  by  Euler  and  Legendre 
for  S2,  S3,  etc.;  it  is  in  fact  the  only  practicable  method  for  small  values  of  «, 
owing  to  the  slow  convergence  of  the  series.  See  Boole's  Finite  Differences,  ad 
Edition,  p.  93. 


§  XXIL]        THE  LOGARITHMIC   DERIVATIVE  OF    F(x).  393 

It  follows  that  if  r  is  a  positive  integer, 


This  equation  may  also  be  derived  by  successive  steps  from  the 
logarithmic  differentiation  of  F(x  +  i)=xF(x).  It  shows  that  the 
sum  of  a  finite  number  of  terms  of  any  harmonic  series  can  be 
expressed  in  terms  of  ^--functions,  that  is  in  terms  of  F-  and  F'- 
functions. 

Putting  x  =  i  in  equation  (i),  or  x  =  o  in  the  derivative  of 
equation  (i),  Art.  360,  we  find  F'(i)  =—•)-.  Hence  from  equa- 
tion (2)  we  have,  when  r  is  a  positive  integer, 


and,  since  F(r+i)=r\, 


In  particular  r'(i)=—  7-,  and  F'(2)  =  i  —  f,  these  give  the 
slope  of  the  graph  of  F(x),  p.  378,  at  the  points  A  and  B.  Again, 
when  r  is  very  great,  we  have  approximately,  from  equation  (i) 
and  the  definition  of  j,  F'(r)/F(r)=logr,  which  shows  that  the 
subtangent  of  this  curve  tends  to  a  ratio  of  equality  to  the  Napier- 
ian logarithm  of  the  abscissa. 

364.  An  important  theorem  due  to  Gauss  follows  readily  from 
equation  (i);  for,  putting  mx  in  place  of  x,  where  m  is  an  integer, 
it  becomes 


394  DEFINITE  INTEGRALS.  [Art.  364. 


d  log  r(mx) 

— 


r,          i        i  i      n 

=    log»  --  ------- 

mx    mx  +  i  mx  +  n  —  iJM=oo 


mdx 
Multiplying  by  m,  the  terms  may  be  written  in  groups  thus: 

d  log  F(mx)  iii  i 

•—mlogn  —  -  — 


dx  x  i  2  m  —  i 

x-\ 


mm  m 

ii  i 


x+i  ,  i  2  m  —  i 

m 


X  +  2  I 

X  +  2+  — 

m 


The  number  of  columns  is  m  and  the  number  of  terms  in 
each  is  nf,  where  n  =  mn'  and  n'  is  to  be  made  infinite.  Now, 
since  m  log  n=m  log  n'  +  m  log  m,  we  can  assign  log  n'  to  each  of 
the  m  columns,  and  then  sum  the  column  by  equation  (i).  Thus 
we  have 

d  .  d  d  I       i  \ 

—  log  r(mx)  =  m  log  m  +  —  log  1  (x)  +  —  log  1  \  x-\ —  I 

J/y*        O          ^  /  /7'Y?  //  "Y"  \  YV\  i 

d  /        2  \  d  I       m  —  i\ 

+—  log  r (x+-  +---+-riogr (»+- 

dx  \       m/  dx  \         m   / 

Integrating  this  equation, 

/       I  \ 
log  r(mx)=C  +  mx  log  w  +  log  r"(^)  +  log  F(x  +  —  )  +  •  •  • 

\  rft/ 

/      m  —  i\ 

+iog  r (x+-    -  , 

X         m  /' 


§  XXII.]         THE   LOGARITHMIC   DERIVATIVE   OF   r(x).  395 

and  taking  the  exponential  of  each  side, 

(i  \     /        2  \  /       m  —  i  \ 

x-t  -  )r(x+-  ...  r(x+-    -),    (7) 
ml     \      ml  \         m    / 


where  A  is  a  constant  of  integration  to  be  determined. 
365.  For  this  purpose,  we  notice  that,  putting  x  =  i/m, 


Now,  by  equation  (6),  Art.  346, 

r(I\r(m-I\_   * 

\m/     V    m   I  rS 


sin  - 
m 


and,  taking  in  like  manner  the  products  of  pairs  of  factors  equi- 
distant from  the  ends, 


sin  —sin  2—  •  •  •  sin  (m  —  i)— 

mm  m 


This,  by  equation  (3),  Diff.  Calc.,  Art.  234,  gives 

m—  i 

^    .  .  (5) 


m 


Substituting  in  equation  (4),  we  find 


DEFINITE   INTEGRALS.  [Ex.  XXII. 

and  this  in  equation  (3)  gives  Gauss'  theorem, 


=  r(mx)(27t)   2  m*-***.    .    (6) 
Equation  (5)  is  a  special  case  of  this  equation. 

Examples  XXII. 
i.  Show  that 


member  is  the  original  form  in  which  Elder  introduced  the 
integral,  p,  q,  and  n  being  integers.  Legendre  thus  reduced  all  cases  to 
that  in  which  «=i,  by  introducing  fractional  values  of  the  arguments. 

2.  Prove  that 

f1*n-I+#m-1,         f1  xn-l+xm-1  . 

J0  (I+xr+»dx=  Ja  (i+xr+n     B(m'n}' 

3.  Prove  that 


4.  From  example  3  derive  the  general  property  of  the  T-function 


Supposing  the  values  of  F(x)  from  x=o  to  x=$to  have  been  found, 
those  from  $  to  f  can  be  derived  from  this  formula,  and  thence  by  means 
of  equation  (6),  Art.  346,  all  values  of  P(x). 


§  XXII.]  EXAMPLES.  397 

5.  Derive  the  continued  product 

l/»2 


r     mn  r        m2   -i  r        w2 

iio"!1  w2J  L1  (»+i)2jL    (w+2 


6.  Show  that 


7.  Prove  that 

B(m,  ri)B(m+n,  l)  =  B(m,  l)B(m  +  l,  ri). 

8.  Derive  the  fundamental  property  of  the  /""-function  by  differen- 

,-co 

tiation  of       xn~  le~axdx  with  respect  to  a,  and  also  by  integration  with 

J  o 

respect  to  a. 

Reduce  the  following  integrals  to  /""-functions : 

11      dz 


10. 


ii 


-TIT — ;dx.  3.496 

J  0|/(an  x) 

:k 


f  2  sin  2p    ld  cos29    ld  dd 
I3-  J  0  (a  sin2  6  +  b  cos2 19)^+9'  aaf^r(p  +  qY 


f°°  i 

e~am**d3C. 

j 0  ma    \ m 

F(»-i) 


398  DEFINITE  INTEGRALS.  [Ex.  XXII. 


16.  I  xl-l(i-x2)m-*dx. 

J  o 

f°° 

17.  cos(bxn)dx, 

J  o 


cos  — . 


18.  Deduce  equation  (3),  Art.   350,  from  (6),  Art.  346,  by  using 
the  form  of  F  given  in  equation  (2),  Art.  348. 

19.  Show  that,  when  «=o,   (/?"-a")r(«)  =  log  A 

20.  Find  the  area  in  the  first  quadrant  enclosed  by  the  curve 

xn      V* 


and  apply  to  the  cases  w=4,  n=  oo  ,  and  n=\. 


ab 
Area=  — 


w=4  gives  A  =  .g2jab;  n=co,A=ab;  n=1[,  A  —  $ab. 

21.  Find  the  volume  generated  by  a  loop  of  the  lemniscate 

r2  =  a2  sin  26 
revolving  about  the  axis  of  x.  i. 233703. 

22.  Find  the  length  of  a  loop  of  the  curve 

ft  =  a*  cos  §6.  3-37490- 

23.  Express  — — -  as  a  definite  z-integral  by  means  of  equation  (2). 

Vx 

/•oo 

/-«/-\C      'V* 

Art.  289,  and  thence  evaluate        — — dx  as  a  double  integral.     Com- 

J  o    V x 
pare  Art.  354.  /n 


§  XXII.]  EXAMPLES.  399 

24.  Show  that  B(m,  ri)  may  be  transformed  into 

(a+b)mbn  Ca  ym~  '(a-  y)n~ x 
aw  +  w-1  J  0       (y  +  b)m+n       y' 

where  a  and  b  are  positive ;  whence 

ra,m-i_)»-i     _    ™+n-* 
M         ~ 


25-  Find  the  value  of  the  multiple  integral 

.  .  .  xl~Iym~Iztt~l  .  .  .  dz  dy  dx 
for  all  positive  values  of  the  variables,  such  that 


p     q     r 
26.  Show  that 


.          3f  3£  ^?  .  I 

i+x     2(2  +  ^)3(3+^)     4(4+^) 


where  5;=a)w+(i)M  +  -..,  etc. 


400  DEFINITE   INTEGRALS.  [Ex.  XXII. 

27.  Prove  that 


and 


—  [ 


28.  Prove  by  direct  summation  of  the  powers  of  the  reciprocals 
that 

52  +  53  +  54  +  554  ----  =1,            and  52-5g  +  54-55-)  ----  =  £. 

29.  Derive  the  numerical  series 


and  thence 

i*»—  i*s+K  ----  =  log  2+7--  1. 
30.  Derive  the  series 

log  ^(i-^2)=      2     ^  ^    ^^+<  m  ^ 
sin  TTJC 

and  thence 

log  2  =  52+i54+^564  ---- 
Hence  also,  from  Ex.  29,  we  may  derive 


31.  Express  r'(x)  as  a  definite  integral,  giving  particular  results 
when  x  =  i,  x=2  and  #=3. 

/•OO  |-00  ,-00 

e~*  logzdz=—  7-;        ze~a  log  2^2=1—7-;        22^~z  log  z  ^2=3  —  2^. 

Jo  Jo  Jo 


§  XXIL] 


VALUES  OF  LOG 


401 


LOG  r(n). 


n 

o 

i 

2 

3 

4 

s 

6 

7 

8 

9 

1  .00 

3. 

9-99975 

95° 

925 

900 

876 

851 

826 

802 

777 

I.  01 

9-99753 

729 

704 

680 

656 

632 

608 

584 

560 

536 

i  .02 

5J3 

489 

466 

442 

419 

395 

372 

349 

326 

3°3 

1.03 

280 

257 

234 

211 

1  88 

166 

H3 

121 

098 

076 

1.04 

°53 

031 

009 

*987 

*965 

*943 

*92I 

*899 

*877 

*855 

i-°5 

^.98834 

812 

791 

769 

748 

727 

705 

684 

663 

^42 

1  .06 

621 

600 

579 

558 

538 

51? 

496 

476 

455 

435 

1.07 

4i5 

394 

374 

354 

334 

3*4 

294 

274 

254 

234 

i.  08 

215 

195 

i75 

156 

137 

117 

098 

079 

°59 

040 

1.09 

021 

002 

*983 

#964 

*946 

*927 

*9o8 

#890 

*87i 

*853 

I.  10 

9  -97834 

816 

797 

779 

761 

743 

725 

707 

689 

671 

1  .11 

653 

635 

618 

600 

582 

565 

548 

53° 

5*3 

496 

I.  12 

478 

461 

444 

427 

410 

393 

376 

360 

343 

326 

I-I3 

310 

293 

277 

260 

244 

228 

211 

i95 

179 

163 

I.I4 

147 

I31 

US 

099 

083 

068 

052 

036 

O2I 

005 

!-I5 

3.96990 

975 

959 

944 

929 

914 

899 

884 

869 

854 

1.16 

839 

824 

810 

795 

780 

766 

751 

737 

722 

708 

i  '.17 

694 

680 

666 

651 

637 

623 

610 

596 

582 

568 

1.18 

554 

54i 

527 

5M 

500 

487 

473 

460 

447 

434 

1.19 

421 

407 

394 

381 

369 

356 

343 

33° 

3i7 

3°5 

i  .20 

292 

280 

267 

255 

242 

230 

218 

206 

194 

181 

I  .  21 

169 

i57 

146 

134 

122 

no 

098 

087 

075 

064 

I  .22 

052 

041 

029 

018 

007 

*995 

*984 

*973 

*g62 

*95i 

1.23 

9-95940 

929 

918 

908 

897 

886 

876 

865 

854 

844 

1.24 

833 

823 

813 

803 

792 

782 

772 

762 

752 

742 

1-25 

732 

722 

712 

7°3 

693 

683 

674 

664 

655. 

645 

I  .26 

636 

627 

617 

608 

599 

59° 

58i. 

572 

563 

554 

1.27 

545 

536 

527 

5i9 

5i° 

5°i 

493 

484 

476 

467 

1.28 

459 

45  1 

442 

434 

426 

418 

410 

402 

394 

386 

I  .  29 

378 

37° 

362 

355 

347 

339 

332 

324 

3i7 

3°9 

I.30 

302 

295 

287 

280 

273 

266 

259 

2^2 

245 

238 

I-3I 

231 

224 

217 

211 

204 

197 

191 

184 

178 

171 

1.32 

165 

159 

152 

146 

140 

134 

127 

121 

"S 

109 

!-33 

10^ 

098 

092 

086 

080 

°75 

069 

063 

058 

052 

i-34 

047 

042 

036 

03I 

026 

020 

OI5 

OIO 

005 

ooo 

i-35 

9  •  94995 

990 

985 

981 

976 

971 

966 

962 

957 

953 

1.36 

948 

944 

939 

935 

93° 

926 

922 

Ql8 

014 

910 

r-37 

90S 

901 

898 

894 

890 

886 

882 

878 

875 

871 

1.38 

868 

864 

861 

857 

854 

8^0 

847 

844 

840 

837 

i-39 

834 

831 

828 

825 

822 

819 

816 

813 

811 

8oF 

402 


DEFINITE   INTEGRALS. 


LOG 


—  (Continued). 


n 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

1  .40 

;.  94805 

803 

800 

797 

795 

793 

790 

788 

785 

7^3 

1.41 

781 

779 

776 

774 

772 

770 

768 

766 

764 

7^3 

1.42 

761 

759 

757 

756 

754 

752 

751 

749 

748 

747 

i-43 

745 

744 

743 

74i 

740 

739 

738 

737 

736 

735 

1-44 

734 

733 

732 

73i 

73° 

73° 

729 

728 

728 

727 

I-45 

727 

726 

726 

725 

725 

725 

725 

724 

724 

724 

1.46 

724 

724 

724 

724 

724 

724 

724 

725 

725 

725 

1-47 

725 

726 

726 

727 

727 

728 

728 

729 

73° 

73° 

1.48 

73  1 

732 

733 

733 

734 

735 

736 

737 

738 

740 

1-49 

74i 

742 

743 

744 

746. 

747 

748 

75° 

75i 

753 

I  .  "O 

754 

756 

758 

759 

761 

763 

765 

767 

768 

770 

1.51 

772 

774 

776 

779 

78i 

783 

785 

787 

790 

792 

1.52 

794 

797 

799 

802 

804 

807 

809 

812 

815 

817 

i-53 

820 

823 

826 

829 

832 

835 

838 

841 

844 

847 

1-54 

850 

853 

856 

860 

863 

866 

870 

873 

877 

880 

i.  55 

884 

887 

891 

895 

898 

902 

906 

910 

914 

917 

1.56 

921 

925 

929 

933 

938 

942 

946 

950 

954 

959 

i-57 

963 

967 

972 

976 

981 

985' 

990 

994 

999 

*oo4 

1.58 

1  •  95°°8 

013 

018 

023 

027 

032 

°37 

042 

047 

052 

J-59 

057 

062 

068 

073 

078 

083 

089 

094 

099 

i°5 

i.  60 

no 

116 

121 

127 

132 

138 

144 

149 

155 

161 

1.61 

167 

173 

179 

185 

190 

196 

203 

209 

215 

221 

1.62 

227 

233 

240 

246 

252 

259 

265 

271 

278 

285 

1.63 

291 

298 

3°4 

3" 

3i8 

324 

33i 

338 

34$ 

352 

1.64 

359 

366 

373 

380 

387 

394 

401 

408 

4i5 

423 

1.65 

43° 

437 

445 

452 

459 

467 

474 

482 

489 

497 

1.66 

505 

Si2 

520 

528 

S36 

543 

55i 

559 

567 

575 

1.67 

583 

59i 

599 

607 

615 

624 

632 

640 

648 

657 

1.68 

665 

673 

682 

690 

699 

707 

716 

724 

733 

742 

1.69 

75° 

759 

768 

777 

785 

794 

803 

812 

821 

830 

1.70 

839 

848 

857 

866 

876 

885 

894 

9°3 

9i3 

922 

1.71 

93  1 

941 

950 

960 

969 

979 

988 

998 

*oo8 

*oi7 

1.72 

9.96027 

°37 

047 

056 

066 

076 

086 

096 

106 

116 

1-7.3 

126 

136 

146 

157 

167 

177 

187 

198 

208 

218 

1.74 

229 

239 

250 

260 

271 

281 

292 

302 

313 

324 

i-?^ 

335 

345 

356 

367 

378 

389 

400 

411 

422 

433 

1.76 

444 

455 

466 

477 

488 

499 

Sii 

522 

533 

545 

1.7-7 

556 

567 

579 

59° 

602 

614 

625 

637 

648 

660 

i.jS 

672 

684 

695 

707 

719 

73i 

743 

755 

767 

779 

1.70 

791 

803 

815 

827 

839 

851 

864 

876 

888 

901 

§  XXII.] 


VALUES   OF 


403 


LOG   F(n}—  (Continued). 


n 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

1.  80 

9.96913 

925 

93S 

950 

963 

975 

988 

*ooo 

*oi3 

*026 

1.81 

9.97038 

°5i 

064 

076 

089 

IO2 

IJ5 

128 

141 

J54 

1.82 

167 

1  80 

r93 

206 

219 

232 

245 

259 

272 

285 

1.83 

298 

312 

325 

339 

352 

366 

379 

393 

406 

420 

1.84 

433 

447 

461 

474 

488 

502 

5i6 

53° 

543 

557 

1.85 

S7i 

585 

599 

613 

627 

641 

656 

670 

684 

698 

1.86 

712 

727 

74i 

755 

770 

784 

798 

813 

827 

842 

1.87 

856 

871 

886 

900 

9i5 

93° 

944 

959 

974 

989 

1.88 

9  .  98004 

018 

°33 

048 

063 

078 

°93 

1  08 

123 

139 

1.89 

154 

169 

184 

199 

215 

230 

245 

261 

276 

291 

i  .90 

3°7 

322 

338 

353 

369 

385 

400 

416 

432 

447 

1.91 

463 

479 

495 

5" 

526 

542 

558 

574 

59° 

606 

i  .92 

622 

638 

654 

671 

687 

703 

719 

735 

752 

768 

J-93 

784 

80  1 

817 

834 

850 

867 

883 

900 

916 

933 

1.94 

949 

966 

983 

999 

*oi6 

*033 

*°5° 

#067 

*o83 

*IOO 

1.95 

9.99117 

134 

151 

1  68 

185 

202 

219 

237 

254 

271 

i  .96 

288 

3°5 

323 

340 

357 

375 

392 

409 

427 

444 

1.97 

462 

479 

497 

5i5 

532 

55° 

567 

585 

603 

621 

1.98 

638 

656 

674 

692 

710 

728 

746 

764 

782 

800 

1.99 

818 

836 

854 

872 

890 

909 

927 

945 

963 

982 

VALUES   OF  S»= 


n 

sn 

n 

sn 

2 

0.64493  40668  5 

14 

0.00006  12481  4 

3 

.20205  69031  6 

15 

.00003  05882  4 

4 

.08232  32337  I 

16 

.00001  52822  6 

5 

.03692  77551  4 

17 

.00000  76372  o 

6 

.01734306198 

18 

.00000  38172  9 

7 

.00834927738 

19 

.00000  19082  I 

8 

.00407  73562  o 

20 

.00000  09539  6 

9 

.00200  83928  3 

21 

.00000  04769  3 

10 

.00099  45751  3 

22 

.00000  02384  5 

n 

.00049  41886  o 

23 

.00000  01192  2 

12 

.00024  60865  5 

24 

.00000  00596  i 

13 

.00012  27133  5 

25 

.00000  00298  o 

N.B.  For  greater  values  of  w,  divide  s^  successively  by  2. 


404  FUNCTIONS   OF    THE  COMPLEX    VARIABLE.     [Art.  366. 

CHAPTER   VI. 
INTEGRATION    OF    FUNCTIONS    OF    THE    COMPLEX    VARIABLE. 


XXIII. 

>v 

Complex  Values  of  the  Derivative. 

366.  In  treating  of  the  functions  of  a  complex  variable  it  is 
usual  to  put 

z  =  x+iy 

for  the  independent  variable;  then  if  iv=f(z)  it  will  usually 
be  of  the  form  w  =  u+iv,  in  which  u  and  v  are  real  quantities 
Involving  the  real  quantities  x  and  y.  It  is  explained  in  Diff. 
Calc.,  Art.  223,  how  a  value  of  z  is  geometrically  represented, 
in  a  plane  of  reference,  by  the  point  whose  rectangular  coordinates 
are  x  and  y,  and  also  that  z  may  be  put  in  the  form  reie,  where 
r  and  6  are  the  polar  coordinates  of  the  same  point,  which  we 
shall  call  the  point  z.  In  this  last  form  r  is  called  the  modulus 
of  z  and  is  regarded  as  its  absolute  value,  while  6  is  the  argu- 
ment or  angle  determining  the  direction  of  the  unit  factor  eiet 
which  is  one  of  the  radii  of  the  unit  circle. 
In  like  manner,  we  may  write 

•w  =f(z)  =  u+iv  = pe^ 

and  represent  w  by  a  point  referred  to  rectangular  axes  of  u 
and  v,  generally  taken  for  convenience  in  another  plane  which 
we  may  call  the  w-plane. 


§  XXIII.]     COMPLEX   VALUES  OF   THE  DERIVATIVE. 


405 


Thus  the  functional  relation  w=f(z)  establishes  a  corre- 
spondence between  positions,  PI,  P2,  P3  etc.,  of  the  point  (x,  y) 
in  the  -z-plane  and  positions,  Ql}  Q2,  Qs  etc.,  of  the  point  (M,  v) 
in  the  w-plane.  Thus  the  w-plane  becomes,  as  it  were,  a  map 
of  the  z-plane,  in  which  any  figure  described  in  the  z-plane  is 
represented  by  a  figure  which  is  called  its  image.  The  mode  in 
which  this  mapping  takes  place  constitutes  the  geometric  repre- 
sentation of  the  function  which  gives  rise  to  it,  just  as  in  the 
simpler  case  of  the  function  of  a  real  variable  the  curve  y  =f(x) 
represents  the  function. 

367.  A  continuous  variation  of  the  independent  variable  z 
is  now  represented  by  a  continuous  motion  of  the  point  z,  which 
•may  be  along  any  arbitrary  path  or  track.  The  point  w  will 
then  describe  a  corresponding  track  in  the  w-plane,  which  is 
the  image  of  the  track  of  z.  Let  z,  represented  by  P  in  Fig.  64, 


u 


FIG.  64. 

describe  its  track  in  a  definite  manner.  Its  motion  at  any  instant 
involves  a  definite  rate  and  a  definite  direction.  Accordingly 
its  differential, 

dz=dx+idy, 

depends  upon  two  independent  arbitrary  elements,  the  values  of 
dx  and  dy,  which  measure  in  fact  the  resolved  velocities  of  P 


406          FUNCTIONS   OF    THE   COMPLEX    VARIABLE.      [Art.  367. 

in  the  directions  of  the  two  axes.     The  actual  velocity,  or  abso 
lute  rate  of  z,  is  measured  by  ds,  the  differential  of  the  arc  described. 
Denoting,  as  usual,  by  $  the  inclination  of  the  curve  to  the  axis 
of  x,  we  have 

dz=dx+idy=dse^  .......     (i) 

In  the  last  member,  the  rate  and  the  direction  of  motion 
are  expressed  separately  by  the  values  of  the  independent  quan- 
tities ds  and  <£;  ds  =  y(dx2  +  dy2)  being  the  modulus  of  dz,  and 
<£  its  argument. 

368.  In  like  manner,  we  have,  for  the  differential  of  the  func- 
tion, 

div=du+i  dv  =  ds'e^',       .....     (2) 

dsf  in  Fig.  64  being  the  absolute  length  or  modulus  of  div,  and  </>' 
its  inclination  to  the  axis  of  u.  From  equations  (i)  and  (2) 
we  obtain  the  derivative  of  the  function  w  =  f(z),  namely 

d-w    du+idv    ds'          , 


=  —  =  -  =  —  -} 

J  (  J     dz     dx+idy    ds  13J 

The  final  member  of  this  equation  shows  that  the  derivative 
is  a  complex  quantity,  of  which  the  modulus  is  the  ratio  of  the 
rates,  and  the  argument  is  the  difference  of  the  directions  of  the 
motions  of  w  and  z.  Since  the  derivative  has  in  general  a  defi- 
nite value  for  a  given  value  of  z,  both  this  ratio  of  rates  and 
this  difference  of  direction  are  independent  of  the  arbitrary  path 
of  z.  Thus,  if  two  paths  of  z  start  from  the  same  point  P,  Fig.  65, 
making  a  given  angle,  (j>\  —  <j>2  =  a,  their  images  in  the  w-plane 
will  make  the  same  angle  at  Q,  that  is,  $\-4>2=(x- 

369.  It  is  an  obvious  consequence  of  this  preservation  of 
angles  in  the  image  that  the  image  of  a  small  area  \vill  be  a  simi- 
lar small  area  (the  ratio  of  similitude  being  the  ratio  of  rates, 
which,  as  mentioned  above,  is  the  modulus  of  the  derivative). 


§  XXIII.] 


CON  FORMAL  REPRESEN  TA  TION. 


407 


Fr  this  reason,  the  image  is  said  to  constitute  a  conformal  repre- 
sentation.* 

Assuming  the  derivative  to  be  finite  and  continuous,  the  scale 
of  representation,  or  magnification,  and  the  orientation  change 
continuously  from  point  to  point.  The  points  for  which  the 


FIG.  65. 

modulus  of  the  derivative  is  zero  or  infinite  (and  its  argument 
therefore  indeterminate)  are  points  of  discontinuity  for  the 
function,  and  at  these  points  the  conformal  representation  fails. 

Conjugate  Functions  of  x  and  y. 

370.  When  the  function  iv=f(z)=f(x-\-iy}  is  given  in  the 
form  u+iv,  u  and  v  become  known  functions  of  the  two  real 
variables  x  and  y.  For  example,  if  f(z)  =  z2, 


—  y2 


whence 


'-y2, 


*  In  like  manner,  a  map  of  a  spherical  surface  in  which  angles  are  preserved 
is  a  conformal  representation  of  the  surface.  The  stereographic  projection  is 
an  example.  In  that  case,  the  magnification  has  at  every  point  of  the  circumference 
of  the  primitive  circle  a  value  double  that  at  the  centre. 


408  FUNCTIONS   OF    THE   COMPLEX    VARIABLE.     [Art.  370. 

In  the  general  equation  expressing  the  derivative,  Art.  368, 
....        du+idv 


the  differentials  du  and  dv  are  total  differentials  due  to  the  varia- 
tion both  of  x  and  of  y.     Thus,  by  Diff.  Calc.,  Art.  370, 

du  j       du  .  dv  .       dv  , 

du  =  —dx+—dy,       dv=  —  dx+—  dy, 
dx         dy   '  dx         dy   ' 

du   du 

in  which  —  ,  —  etc.  are  the  partial  derivatives  of  the  functions 
dx  dy 

u  and  'v,  while  dx  and  dy  are  independent  differentials.      Substi- 
tuting in  equation  (i), 


du     .  dv  \         (du     .  dv\  , 
—  +*  —  \dx+{—+i—)dy 

dx    dx>      \dy    dy' 


dx+idy 

Since  f(z)  has  a  value  independent  of  the  ratio  dy  :  dx,  we  have, 
by  putting  dy  and  dx  successively  equal  to  zero, 

ftt  \  —du  ,  •  dv  _      .du     dv 

J  (2  )        —  +  1  —  1  —  —  +  —  —  . 

dx      dx  dy     dy 

Equating  separately  the  real  and  imaginary  parts  of  the  last 
equation,  we  have  the  following  relations  between  the  partial 
derivatives  : 

du^dv          dv        du 
dx    dy'        dx        dy' 

Again,  eliminating  v  from  these  two  equations,  we  have 

d2u    d2u 


and  the  same  equation  may  be  found  for  v. 


§  XXIII.]     CONJUGATE  FUNCTIONS  OF  X  AND   y.  409 

Two  functions  of  x  and  y  which  together  satisfy  equations  (2) 
are  called  conjugate  functions.  Each  of  them  necessarily  satisfies 
equation  (3).  Thus  the  functions  x2  —  y2  and  2xy,  derived  above 
from  the  function  z2,  are  conjugate  functions  and  will  be  found 
to  satisfy  these  equations.* 

371.  From  the  expressions  for  u  and  v  in  terms  of  x  and  y 
we  can  readily  obtain  the  equation  of  the  image  in  the  i^-plane 
of  a  curve  in  the  z-plane  whose  equation  is  given.  For  we  have 
only  to  eliminate  x  and  y  from  three  given  equations.  For 
example,  in  the  case  cited  above,  where 

u  =  x2—  y^,        v=2xy, 
we  thus  find,  corresponding  to  x  =  a, 


which  is  the  equation  of  a  parabola  with  focus  at  the  origin  and 
vertex  on  the  axis  of  u  to  the  right  of  the  origin.  In  like  manner,. 
we  find  corresponding  to  y  =  b, 


'which  is  a  parabola  with  focus  at  the  origin  and  vertex  on  the 
axis  of  u  to  the  left  of  the  origin. 

If  a1  series  of  equidistant  values  be  given  to  a  and  also  to  b,  we 
shall  have  in  the  z-plane  two  sets  of  straight  lines  parallel  to  the 
axes,  dividing  the  z-plane  into  small  squares.  The  two  corre- 
sponding systems  of  parabolas  in  the  w-plane  cut  each  other 


*  An  expression  u  +  iv  in  which  the  functions  u  and  v  are  taken  at  random 
(although  a  function  of  z  in  the  sense  that  it  is  determined  by  the  position  of  the 
point  2)  is  not,  unless  u  and  v  satisfy  the  equations  above,  an  analytical  function 
of  z,  that  is  one  which  can  result  from  algebraic  operations  performed  upon  z, 
that  is,  upon  x+iy  as  a  whole.  Analytical  functions  were  called  by  Cauchy 
"fonctions  monogenes." 


410          FUNCTIONS   OF    THE  COMPLEX    VARIABLE.      [Art.  371. 

orthogonally,  in  accordance  with  Art.  368,  and  divide  the  w-plane 
into  curvilinear  squares.  The  correspondence  of  these  small 
squares  in  the  two  planes  exhibits  most  clearly  the  mode  in  which 
the  function  w  varies  with  the  independent  variable  z.* 


Two-valued  Functions. 

372.  Let  w  be  a  two-valued  function,  and  let  z  be  repre- 
sented by  the  point  P  moving  along  a  given  track  from  the  initial 
point  z0  to  the  final  point  z\.  Then  the  two  corresponding  points 
in  the  w-plane  will  in  general  be  distinct  points,  Q  and  Q',  and 
will  describe  tracks,  say  AB  and  A'B',  which  have  no  common 
point;  that  is  to  say,  no  point  at  which  Q  and  Q'  arrive  simul- 
taneously. Hence,  selecting  A  as  the  initial  value  of  w,  B  will  in 
general  be  determined  without  ambiguity  as  the  final  value  of  w. 
But,  if  the  track  of  z  passes  through  a  point  for  which  the  two 
values  of  w  become  equal,  the.  paths  AB  and  A'B'  will  have  a 
common  point,  and  there  will  be  an  ambiguity  in  the  final  value 
of  w,  because  it  will  be  possible  to  pass  from  A  to  B'.  Such  a 
point  in  the  z-plane  is  called  a  change-point  or  branch-point  for 
the  given  function. f  Then,  provided  it  does  not  pass  through  a 
branch-point,  the  track  of  z  (together  with  the  initial  value  w0) 
determines  without  ambiguity  the  final  value  w\. 


*  Figures  illustrating  in  this  way  many  of  the  elementary  functions  will  be 
found  in  Dr.  Thomas  S.  Fiske's  Functions  of  a  Complex  Variable,  p.  13  et  seq. 
(Mathematical  Monograph  Series,  John  Wiley  &  Sons,  1906),  to  which  the 
reader  is  referred  for  an  excellent  introduction  to  the  Theory  of  Functions. 

It  will  be  noticed  that,  in  the  case  above,  illustrating  the  function  z2,  the  whole 
of  the  w-plane  corresponds  to  one-half  of  the  2-plane.  On  the  other  hand,  only 
a  portion  of  the  w-plane  will  correspond  to  a  single  value  of  a  multiple-valued 
function. 

t  Two  values  of  w  may  become  simultaneously  infinite  at  a  branch-point,  in 
which  case  the  conclusions  below  may  be  established  by  means  of  the  function 
•v/~l,  of  which  two  values  become  zero. 


§  XXIII.]  TWO-VALUED   FUNCTIONS.  411 

373.  Moreover,  if  the  track  of  z  be  altered  gradually,  that  is 
by  continuous  or  infinitesimal  changes,  to  any  other  track  between 
z0  and  Zi,  the  tracks  AB  and  A'B'  in  the  w-plane  will  suffer  con- 
tr  uous  change,  but  it  will  be  impossible  to  pass  from  A  to  B' 
unless  at  some  instant  these  tracks  have  a  common  point;  that 
is,  unless  the  track  of  z  passes  through  a  branch-point.  It  follows 
that  a  track  which  can  be  altered  into  a  given  track  without  ever 
passing  through  a  branch-point  will  determine  the  same  final 
value  of  w.  Such  tracks  are  said  to  be  reducible,  one  to  the 
other.  This  is  as  much  as  to  say  that,  if  the  area  enclosed  by 
the  two  tracks  of  z  does  not  contain  a  branch-point,  the  tracks 
lead  to  the  same  final  value  of  w;  so  that,  if  z  returns  to  z0  after 
describing  the  complete  contour  formed  by  the  two  tracks,  it  will 
return  with  the  same  value  of  w  with  which  it  started.  But,  if 
the  contour  encloses  a  branch-point,  it  may  return  with  the  other 
value  of  w. 

374-.  Take,  for  example,  the  simple  function  w=\fz.  Put- 
ting z  in  the  form  rei0,  we  have  seen  in  Diff .  Calc.,  Art.  228,  that 
w  admits  of  the  two  values  re*ie  and  ^re^e+ajc)  which  is 
equal  to  —  yre^e.  As  z  moves,  the  argument  0  varies  con- 
tinuously. Now,  starting  with  the  initial  values  z0  =  r0eio°, 
iv0=  ^r0e^e°,  if  z  returns  to  z0  without  encircling  the  origin, 
6  will  return  to  60,  and  w  to  w0;  but,  if  z  encircles  the  origin  in 
the  positive  direction,  the  final  value  of  the  argument  of  z  will 
be  d0+27t,  and  the  final  value  of  w  will  be  —  w0.  If  z  describes 
the  same  contour  in  the  negative  direction,  the  final  value  will 
again  be  —  w0;  but,  if  it  enwraps  the  origin  twice,  or  any  even 
number  of  times,  w  will  return  to  the  value  w0. 

375.  In  like  manner,  if  w—^(z—a).^>(z),  a  is  a  branch- 
point, and  a  circuit  about  a  described  by  z  will  change  the  sign 
of  w,  provided  the  factor  <f>(z)  has  no  branch-point  either  on  or 
within  the  circuit  described  by  z.  Thus  the  function 

—  a)(z  —  b)\ 


412  FUNCTIONS   OF    THE   COMPLEX    VARIABLE.     [Art.  375. 

is  two-valued,  and  has  two  branch-points,  a  and  b.  If  z  describes 
a  contour  enclosing  a,  but  not  enclosing  6,  the  sign  of  w  will 
be  changed;  and  so  also  if  the  contour  encloses  b  and  not  a. 
But,  if  the  contour  encloses  both  a  and  b,  w  will-  return  to  the 
value  w0  when  z  returns  to  z0,  so  that  the  function  is  virtually 
one-valued  for  such  a  contour.  Since  the  direction  in  which  z 
moves  about  either  of  these  branch-points  is  indifferent,  the  same 
thing  will  be  true  if  the  track  of  z  forms  a  figure-of-eight,  of 
which  one  loop  contains  a  and  the  other  b. 

Multiple-valued  Functions. 

376.  The  conclusions  arrived  at  in  Arts.  372  and  373  apply 
also  to  functions  having  more  than  two  values.  Thus,  we  select 
an  initial  value  z0,  to  this  corresponds  a  number  of  values  of  w0. 
Then,  any  track  (avoiding  branch-points)  from  the  initial  point 
to  the  point  Zi  leads  from  any  one  of  the  values  of  w0  to  a  defi- 
nite one  of  the  values  of  w\.  Now,  choosing  one  of  the  values 
of  w0  as  the  initial  value,  different  tracks  may  lead  to  different 
values  of  w\\  but,  if  the  area  between  two  tracks  does  not  con- 
tain a  branch-point,  the  values  reached  are  the  same. 

A  standard  track  between  z0  and  Z1}  for  example  the  rectilinear 
one,  will  establish  a  correspondence  between  the  values  of  w0 
and  those  of  w\\  then  one  value  of  w0  and  the  corresponding 
value  of  iv\  are  selected  as  primary  values.  Now,  any  track  may, 
without  passing  through  a  branch-point,  be  reduced  either  to  the 
standard  track  or  to  one  combining  with  it  one  or  more  loops 
or  contours  (from  and  back  to  the  initial  point),  each  surrounding 
a  single  one  of  the  branch-points.  The  result  of  describing  one 
of  these  loops  is  to  pass  from  one  to  another  of  the  values  of  w0, 
and  thus,  in  connection  with  the  standard  track  from  z0  to  z\} 
we  may  pass  from  the  primary  value  cf  w0  to  one  of  the  non- 
primary  values  of  w\. 

377.  For    example,  let    the    function  be  w=tyz,  for  which 


§  XXIII.]  MULTIPLE-VALUED   FUNCTIONS.  413 

the  origin  is  the  only  branch-point.  Take  unity,  represented 
by  the  point  A  in  Fig.  66,  as  the  initial  value  of  z,  and  let  B  rep- 
resent the  final  value  of  z.  Putting  z=reio,  we  take  zero  for 
the  initial  value  of  6.  Then  the  value  of  w  resulting  from  a 
rectilinear  track  will  be  tyre*ie,  where  6  must  lie  between  —  - 
and  7t  in  value.  Call  this  the  primary  value  of  w,  so  that  the 
primary  value  of  w  at  A  is  unity,  and  at  B  it  is  iv\  =  f/rie*^'. 

The  result  of  making  the  circuit  of  the  origin  in  the  posi- 
tive direction  is  to  add  2it  to  the  value  of  6,  and  therefore  to 
multiply  w  by  e&*=io,  oie  of  the  r 

cube  roots  of  unity.  (DifL  Calc., 
Art.  230.)  Thus,  when  the  track 
of  z  is  ACB  in  Fig.  66,  the  initial 
value  of  w  being  unity,  its  final 
value  is  urw\.  Again,  the  result 
of  making  the  circuit  of  the 
origin  in  the  negative  direction 
is  to  multiply  w  by  a/*,  the  other 
cube  root  of  unity;  so  that,  when  the  track  of  z  is  ADC,  the  re- 
sult is  uf"w\.  This  last  is  also  the  result  of  a  track  enwrapping 
the  origin  twice  in  the  positive  direction. 

378.  The  function  iv  =  logz  presents  a  branch-point  of  a 
different  character.  We  have,  on  putting 


FIG.  66. 


z  =  re 


w  =  log  z  =Log 


where  Log  r  is  the  real  logarithm  of  the  modulus.  Taking  unity 
(with  6  =  0)  for  the  initial  value  of  z,  the  initial  value  of  w  is 
zero.  Let  us  denote  by  the  symbol  Log  z  the  result  when  z  follows 
the  rectilinear  track  AB  in  Fig.  66.  Then  Log  z=Logr+«00, 
where  60  lies  between  n  and  —TI.  The  circuit  of  the  origin  in 
the  positive  direction  adds  271  to  6,  and  therefore  adds  2in  to  w. 
Thus,  when  the  track  of  z  is  ACB,  the  final  value  of  w  will  be 
z  +  2wr,  and  for  the  track  ADB  it  is  Log  z—  2«r,  while  the 


414         FUNCTIONS   OF    THE   COMPLEX    VARIABLE.      [Art.  378. 

general  value  log  z=Log  r+i00+2nix  is  the  result  of  n  circuits 
of  the  origin  combined  with  the  rectilinear  track  AB. 

379.  The  values  selected  as  primary  are  said  to  constitute 
one  branch  of  the  function,  but  the  mode  of  selection  is  arbitrary. 
A  given  selection  corresponds  to  a  cut  in  the  z-plane,  or  a  number 
of  cuts,  each  starting  from  a  branch-point,  and  either  passing  to 
infinity  or  terminating  in  another  branch-point.  Moreover  every 
branch-point  must  be  one  of  the  extremities  cf  a  cut.  Starting 
from  any  initial  point,  it  is  possible  to  reach  any  other  point  of 
the  plane  without  crossing  a  cut,  and  the  values  of  w  so  obtained 
constitute  the  branch.  For  example,  the  selection  above  for  the 
primary  value  of  the  logarithm  is  equivalent  to  making  a  cut 
along  the  negative  part  of  the  real  axis. 

A  Riemanri's  Surface  consists  of  a  number  of  leaves  identical 
with  the  cut  plane  and  joined  to  one  another  along  the  cut,  so 
that  when  we  cross  the  cut  we  pass  into  another  branch  of  the 
function.  In  the  case  of  the  logarithm,  the  number  of  leaves  is 
infinite,  and  we  ascend,  as  it  were,  from  leaf  to  leaf  as  we  go  round 
the  branch-point  (Windungspunkt  in  German)  in  one  direction. 
For  the  function  in  Art.  375,  the  cut  may  be  made  along  the  line 
ab.  There  are  in  that  case  but  two  leaves,  and  each  is  joined  to 
the  other  along  the  cut,  so  that  the  surface  there  intersects  itself. 

A  multiple-valued  function  may  be  regarded  as  one-valued 
on  its  Riemann's  Surface;  that  is  to  say,  to  each  point  of  the 
surface  there  corresponds  a  single  value  of  the  function. 

Meaning  of  Integration  when  the  Variable  is  Complex. 

380.  Let  us  now  consider  the  meaning  of  an  integral  \vhen 
z  in  the  expression,  /(z)cte,  under  the  integral  sign  admits  of 
complex  values.  Supposing  F(z)  to  be  a  function  such  that 
its  derivative  F'(z)=f(z),  we  write 

F(*)= /(*>&, 


§  XXIII.]     INTEGRATION   OF  COMPLEX  FUNCTIONS.  415 

in  which  /(z)  is  sometimes  called  the  integrand,  while  F(zJ  is  the 
indefinite  integral.  To  remove  the  ir.definiteness  due  to  the 
constant  of  integration  we  write  also,  as  in  Art.  82, 

\f(z)dz=F(z)-F(a), (i) 

J  a 

in  which  a  is  a  selected  initial  value  of  z,  and  it  is  understood 
that  z  in  the  integral  varies  continuously  from  the  initial  value 
a  to  the  final  value  denoted  by  z  in  the  second  member.  To 
the  latter,  we  may  of  course  assign  a  separate  symbol,  and  place 
it  as  the  upper  limit  of  the  integral. 

The  initial  value  a  and  the  final  value  z  in  equation  (i)  may 
now  be  complex  quantities,  and  the  track  described  by  the  cur- 
rent variable  z,  or  track  of  integration,  is  supposed  to  be  knowrn. 

381.  Let  the  indefinite  integral  be  represented  by  a  point 
]V  =  F(z)  in  a  new  plane;  then,  as  z  describes  the  given  track, 
IF  starting  from  the  position  F(a)  moves  in  a  direction  and  with 
a  rate  which  depend  at  every  instant  upon  the  corresponding 
value  of  f(z)dz.  Provided;  therefore,  that,  the  track  of  z  does  not 
pass  through  a  point  for  which /(z)  is  either  infinite  or  ambiguous  * 
in  value,  the  track  of  W  will  terminate  in  a  definite  point  F(b), 


*  It  will  be  noticed  that  the  restriction  with  regard  to  intermediate  points 
upon  the  track  of  z  is  the  same  as  that  of  Art.  82  with  regard  to  the  intermediate 
values  in  the  case  of  real  integrals.  But,  whereas  for  real  integrals  this  restriction 
excluded  the  interpretation  of  certain  integral  expressions,  no  such  exclusion 
now  exists.  For  example,  we  have  when  2  is  a  real  variable 

f'dz  i 


but,  if  the  upper  limit  is  negative,  the  integral  is  inadmissible,  because  to  reach 
such  a  value  z  must  pass  through  the  value  zero  for  which  the  integrand  is  in- 
finite. Whereas,  when  complex  values  are  admitted,  the  track  of  integration 
need  not  pass  through  the  origin  and  the  equation  holds  true.  Thus,  \1  z=  —  ^, 
the  value  of  the  integral  is  2. 


4l6  FUNCTIONS  OF   THE   COMPLEX    VARIABLE.      [Art. '381. 

and,  by  equation  (i),  the  integral  will  have  a  definite  value.  Thus 
the  critical  points  through  which  the  track  of  z  must  not  pass 
include  the  infinity  points,  or  infinities,  of  the  integrand  f(z)  as 
well  as  its  branch-points. 

These  critical  points  are  the  only  positions  at  which  branch- 
points .of  the  indefinite  integral  F(x]  can  occur.  Hence  it  follows, 
as  in  Art.  373,  that  any  two  tracks  which  can  be  reduced  one 
to  the  other  without  passing  through  a  critical  point  will  give 
the  same  value  to  the  integral. 

Integration  around  a   Closed  Contour. 

382.  It  follows  from  the  preceding  article  that,  when  z 
describes  a  closed  curve  or  contour  returning  to  its  initial  value, 
the  integral  must  vanish  unless  the  contour  encloses  one  or  more 
critical  points.  Furthermore,  in  comparing  the  results  of  different 
tracks  of  integration  between  the  same  limiting  points,  we  may 
adopt,  as  in  Art.  376,  a  standard  track,  and  reduce  any  other 
track  to  a  combination  of  this  with  one  or  more  loops  or  con- 
tours, each  enclosing  a  single  critical  point.  For  example,  let 
us  take  the  integral  of  dz/z,  for  which  the  origin  is  an  infinity 
of  the  integrand.  Let 

/.-!** 


denote  the  result  of  integrating  from  an  initial  point  c  in  a  contour 
described  in  the  positive  direction  "about  the  origin  and  back  to  c. 
Putting  c  =  re*e°,  we  may  take  for  the  contour  a  circle  of  radius  r 
and  centre  at  the  origin.  This  is  equivalent  to  putting  z=reie  and 
making  r  constant  while  6  varies  from  60  to  60+2-.  We  have 
now  dz=ireiedd)  hence 

L  =  i 


§  XXIII.]  INTEGRATION  ABOUT  A   POLE.  4!7 

Thus  it  appears  that,  in  this  case,  the  value  of  the  contour 
integral  is  independent  of  the  initial  point  c.  Taking  unity 
as  the  initial  point,  the  value  of  the  indefinite  integral  is  log  z,  and 
2in  is  the  difference  between  consecutive  values  of  this  many- 
valued  function.  Compare  Art.  378. 

Integration  about  a  Pole. 

\ 

383.  If  a  is  an  infinity  of  f(z),  we  shall  assume  that  there 
is  a  positive  value  of  n  such  that  the  product  (z  —  a)nf(z)  has  a 
finite  value  when  z=a.  Denoting  this  product  by  <£(z),  we  have 


where  <j>(z)  is  such  that  0(<z)  is  neither  zero  nor  inirmite.  If 
n  is  an  integer,  a  is  called  a  pole  of  f(z)  of  the  order  n.  For 
example,  tan  z  has  a  pole  of  the  first  order,  or  simple  pole,  at 
|TT;  because  (z—  J?r)  tan  z  has  a  finite  value  when  z=\it.  If 
n  is  fractional,  a  is  a  branch-point  as  well  as  an  infinity  of  f(z). 

We  shall  now  show  that  the  result  of  integrating  f(z)  in  a 
contour  about  a  pole,  not  enclosing  any  other  critical  point,  is  in- 
dependent of  the  initial  point.  In  Fig.  67, 
let  a  be  the  pole,  c  the  initial  point  of  a  con- 
tour cdc  and  b  any  other  point  from  which  • 
a  contour  beb  is  described  about  a  in  the  " 


same    direction.     Since    by   hypothesis    there 
is  no  critical  point  between  the  contours,  the 
contour  cdc  may  be  reduced  without  passing 
through  a  critical  point  to  cbebc,  consisting  of 
the  straight  line  cb  followed  by  the  contour  beb  about  a  returning 
to  &,  and  then  back  to  c  by  the  straight  line.     Since /(z)  has  no 
branch-point  within  the  contour,  it  must  return  to  b  with  the 
/      same  value  with  which  it  commences  to  describe  the  loop  beb, 


418          FUNCTIONS  OF   THE   COMPLEX   VARIABLE.     [Art.  383. 

so  that  the  two  rectilinear  portions  of  the  integral  cancel  each  other. 
Hence  the  two  contour  integrals  must  be  equal.*  In  other 
words,  the  contour  integral  or  loop-integral  is  independent  of  the 
initial  value.  Its  value  may  be  denoted  by  Ia  to  indicate  its 
dependence  upon  the  pole  a. 

384-.  To  obtain  the  value  of  Ia  in  the  case  of  a  simple  pole, 
that  is,  when  n=  i,  we  notice  that  if  (£(z)  were  constant,  we  should 
have,  by  putting  z—a=reie  as  in  Art.  382,  the  result  7a  =  2^0(a). 
But,  by  the  preceding  article,  this  must  also  be  the  result  when 
<£(z)  is  variable,  because  we  can  give  the  circular  contour  in  the 
demonstration  as  small  a  radius  as  we  please. 

The  result  may  also  be  established  as  follows:   The  function 

has  a  finite  value,  namely  <£'(z),  when  z=a,  therefore 

Dntour. 
dz=o; 


2    a 
it  has  no  critical  point  within  the  contour.    Hence,  by  Art.  382, 


Jc       z-a 
therefore 


where  the  suffix  C  indicates  a  contour  of  integration  about  a. 

385.  When  the  pole  is  of  the  second    order,  the  result  of 
putting  z—a=reie  and  making  r  constant  is 

Ia  = 


This  expression,  in  which  r  can  be  diminished  indefinitely,  takes 
the  indeterminate  form  when  r=o  (because  the  integral  of  e~iedd 
between  o  and  27r  vanishes),  and  evaluating  by  the  usual  process 

*  The  rectilinear  portions  of  the  reduced  track  are  separated  by  a  small  space 
in  the  figure  to  show  that  equivalent  contours  must  be  described  in  the  same 
direction. 


§  XXIII.]     INTEGRALS  OF  FUNCTIONS  WITH  POLES.  419 

we  have  Ia=2in  <f>'(a).  This  result  is  also  deducible  from 
equation  (i)  above  by  taking  derivatives  with  respect  to  a;  and, 
in  like  manner,  by  taking  successive  derivatives  we  have  the 
series  of  equations  : 


and,  in  general, 

-'^'W  ......     (5) 


The  contours  are  here  supposed  to  be  described  in  the  positive 
direction,  and,  if  the  pole  is  enwrapped  k  times,  the  result  is 
multiplied  by  k. 

Again,  if  a  function  has  two  poles  a  and  6,  it  is  readily  seen 
that  the  result  of  a   contour  encircling  both  a  and  b  will  be 


Integrals  of  Functions  with  Poles. 

386.  It  follows  from  the  preceding  articles  that  the  integral 
of  a  function  with  poles  will  be  a  function  having  an  unlimited 
number  of  values  differing  by  multiples  of  one  or  more  constants 
dependent  upon  the  poles. 

Take,  for  example,  the  integral 

da 


',-i)(z+i)' 


420  FUNCTIONS  OF   THE  COMPLEX   VARIABLE.    [Art.  386. 

The  poles  of  the  integrand  are  the  points  i  and  —  i.    The  contour 
integral  about  the  first  is,  by  equation  (i), 


— 1 


<— 2«r 

and,  in  like  manner,  we  find  /_t=  --. 

The  initial  point  is  so  taken  that,  for  real  positive  or  negative 
values  of  z,  the  integral  is  the  primary  value  of  tan~Jz,  as  defined 
in  Diff.  Calc.,  Art.  76.  The  rectilinear  track  of  integration  may 
now  be  used  to  define  the  primary  value  of  the  function  for  all 
points  except  those  on  the  imaginary  axis  beyond  the  points  i 
and  —i.  This  selection  of  the  primary  value  therefore  corre- 
sponds to  cuts  of  the  plane  (see  Art.  379)  made  along  these  por- 
tions of  the  axis.  It  follows  that  any  circuit  which  crosses  a 
cut  from  the  right  side  to  the  left  of  the  axis  adds  -  to  the  value 
of  the  function,  and  one  crossing  in  the  reverse  direction  sub- 
tracts TI.  In  particular,  a  circuit  about  both  poles  restores  the 
value  of  the  function. 

Art.  382  and  the  present  one  illustrate  the  fact  that,  when  the 
integrand  has  a  pole,  integration  gives  rise  to  a  function  which 
.admits  of  an  infinite  number  of  values  differing  by  multiples  of 
•a,  constant.  It  follows  that  the  inverse  of  such  a  function  is  a 
periodic  function,  in  which  values  of  the  independent  variable 
.differing  by  multiples  of  a  constant  (called  the  period)  correspond 
to  the  same  value  of  the  function.  Thus  e*  and  tan  z  are  periodic 
functions,  the  first  having  the  pure  imaginary  period  2wr  and 
the  second  the  real  period  n. 

Integration  aboiit  a  Branch-Point. 

387.  The  value  of  a  contour  integral  about  a  branch-point 
of  the  integrand,  unlike  that  about  a  pole,  depends  upon  the  posi- 
tion of  the  initial  point.  The  reason  is  that  when  a  circuit  of  the 


§  XXIII.]    INTEGRATION  ABOUT  A   BRANCH  POINT.  411 

branch-point  is  completed  the  integrand  returns  with  a  new  value; 
thus,  referring  to  Fig.  67,  if  a  were  a  branch-point,  the  integrand 
in  the  contour  cbebc  would  have  a  different  value  when  z  finally 
leaves  b  from  that  with  which  it  first  arrived  at  6,  hence  the  two 
rectilinear  parts  of  the  integral  would  no  longer  cancel  one  another. 
In  illustration,  let  us  integrate  zn  dz  in  a  circle  whose  centre 
is  the  origin,  from  the  initial  point  c  back  to  c.  Except  when  n 
is  an  integer,  the  origin  is  a  branch-point  of  the  integrand.  Put- 
ting c  =  rei°°  and  z  =  reid,  the  contour  is  described  by  making  r 
constant  while  6  varies  from  60  to  60+2n.  Hence  the  contour 
integral  is 

fC  fSo~^~3it  ~n-\-  I  ~"j00+2* 

zn  dz  =  i  Yn ~^~ r  I          c*(n^~I^d0^ gt'i''+i)0 

Jfl  •  H  +  I  Jg 

JO  w    I/O  PO 


n  +  i  n  +  i1 

Now  -   —  is  the  indefinite  integral,  and  e3i(*+l)*  is  one  of 
n  +  i 

the  complex  values  of  i"+I,  hence  the  contour  integral  is  (as 
we  should  expect)  the  quantity  which  must  be  added  to  one 
of  the  multiple  values  of  the  integral  in  order  to  produce  another. 
In  this  example,  if  n  is  zero  or  a  positive  integer,  the  contour 
integral  vanishes,  the  indefinite  integral  being  in  that  case  a  one- 
valued  function.  When  n=  —  i,  the  contour  integral  takes  the 
indeterminate  form,  and  is  found  on  evaluation  to  be  2ix,  agreeing 
with  Art.  382,  and  when  n  is  any  other  negative  integer,  the  contour 
integral  vanishes  in  accordance  with  equation  (5),  Art.  385. 

Integrals  involving  Radicals. 

388.  When  an  integral  involves  an  ordinary  quadratic  radi- 
cal, every  point  at  which  the  quantity  under  the  radical  sign 
vanishes  is  a  branch-point.  If  the  branch-point  is  on  the  real 


422  FUNCTIONS  OF   THE  COMPLEX   VARIABLE.     [Art.  388. 


axis,  and  a  real  initial  value  is  chosen,  the  value  of  the  contour 
integral  is  determined  by  a  real  definite  integral. 
Take  for  example  the  integral 

dz 


-L 


The  critical  points   are    +i    and    —  i.     The  contour  from 
O  about  A,  the  first  of  these  points,  may  be  reduced,  as  in  Fig.  68, 

to  the  rectilinear  track  from 
O  to  a  point  as  near  as  we 
choose  to  A,  followed  by  a 
circular  track  about  A,  and 
t  e  return  to  O  by  the  rec- 
tilinear track.  The  effect 
FIG.  68.  of  the  circular  part  is  to 

change  the  sign  of  the  radical,  so  that  we  may  write 

ri—3         fig  |-o  ^ 

f*-J.    ^7(7^)  +result  of  circular  track+J ,., ^7(fip). 

in  which  the  results  of  the  two  rectilinear  parts  are  equal. 
For  the  circular  part,  put  z=i  —  det9,  so  that  the  circle  is 
described  by  keeping  d  constant  and  varying  6  from  o  to  2x. 
Then  dz=—ideie,  and  the  equation  becomes 


in  which  d  may  be  decreased  without  limit     The  second  term 
vanishes  with  o;  hence,  putting  0  =  0,  we  have 


IA  = 


dz 


(i) 


389.  The  direction  in  which  the  circuit  of  A  is  made  does 
not  affect  the  value  of  I  A,  but  it  must  be  remembered  that  the 


§  XXIII.]         INTEGRALS  INVOLVING  RADICALS.  423 

value  given  in  equation  (i)  implies  that  the  initial  point  is  the 
origin  and  that  the  initial  value  of  the  radical  is  +i.  Since  the 
sign  of  the  radical  is  changed  by  the  circuit,  the  result  of  a 
second  circuit  would  be  —  TT.  That  is,  the  result  of  two  circuits 
about  A  would  be  zero,  and  the  integrand  would  return  to  its 
original  value. 

In  like  manner,  we  shall  find  /#=—  x,  assuming  as  before 
that  the  initial  value  of  the  radical  is  +i.  But,  if  the  circuit 
of  B  is  made  after  the  circuit  of  A  (which  changes  the  sign  of 
the  radical),  the  result  will  be  TT,  and  the  whole  result  of  a 
contour  encircling  first  A  and  then  B,  like  OCO  in  Fig.  68,  will 
be  27T.  In  this  contour  the  integrand  is  virtually  one-valued, 
see  Art.  375,  whence  it  can  be  shown,  exactly  as  in  Art.  383,  that 
the  value  of  the  contour  integral  is  independent  of  the  initial 
point.  Hence,  for  the  contour  which  does  not  pass  through  O, 
as  well  as  for  OCO,  we  may  write  IAB=2x.  The  two  branch- 
points are  thus  together  equivalent  to  a  pole.* 

390.  Let  us  now,  for  any  point  z,  define  the  primary  value 
FI  of  the  integral  V  as  the  result  of  integrating  along  the  recti- 
linear track  Oz  in  Fig.  69.  Consider  now  the  result  of  inte- 
grating along  the  track  OCz.  This  track  may  be  reduced  to 
the  contour  about  A  followed  by  the  track  Oz.  Hence,  remem- 
bering that  the  sign  of  the  integrand  is  changed  by  the  contour, 
the  result  is  n—V\. 

The  results  obtained  in  the  preceding  article  also  show  that 
a  contour  encircling  A  twice  before  reaching  z  will  produce  FI. 

*  Accordingly,  if  2  describes  a  very  large  circle  with  centre  at  the  origin,  the 

integral  approaches  the  form     — R  — ±»|— ,  for  which,  by  Art.  382,  the 

J  V\z  )  J  z 

loop-integral  is  ±2jr.  The  ambiguous  sign  is  due  to  the  fact  that  we  have  not 
determined  which  of  its  two  values  to  give  the  integrand  at  any  one  point  of  the 
circuit.  In  Fig.  68  we  obtained  +  27:  for  the  circuit  OCO,  because  we  assumed 
the  radical  to  have  the  value  + 1  at  O. 


424          FUNCTIONS  OF   THE  COMPLEX   VARIABLE.     [Art.  390. 


One  encircling  B,  like  ODz  in  the  figure,  produces  —n—V\. 
Again,  one  encircling  A  and  then  B  in  either  direction  produces 
27r  +  Vi.  Thus,  by  means  of  different  tracks,  we  can  reach 
any  of  the  values  included  in  the  formulae 

2«7r  +  Fi         and         (2«  +  i)7r-Fi, 

where  n  is  an  integer. 

The  integral  defines  the  function  sin-12  for  complex  values 
of  2,  and  we  have  thus  shown  that  the  relations  which  exist 

between  the  multiple 
values  when  z  is  real 
hold  also  when  z  is 
complex. 

The  fact  that  a 
circuit  of  both  branch- 
points is  equivalent 
to  that  of  a  pole  gives 
a  periodic  character 
to  the  inverse  function,  just  as  in  Art.  386. 

391.  The  use  of  the  rectilinear  track  to  define  the  primary 
branch  of  this  function  is  equivalent  to  making  cuts  in  those 
parts  of  the  real  axis  which  are  beyond  the  points  A  and  B.  The 
Riemann  surface  consists  of  an  infinite  number  of  leaves.  If 
L0,  LI,  L,2,  LS  etc.  denote  consecutive  leaves,  L0  and  LI  may 
be  joined  along  the  positive  cut  so  as  to  form  a  self -intersecting 
surface  (as  in  the  second  illustration  used  in  Art.  379),  and  L2 
and  L3  are  joined  in  like  manner.  But,  along  the  negative  cut 
LI  and  L2  must  be  thus  joined,  and  L3  in  like  manner  with  Z4, 
and  so  on. 

In  Fig.  69,  the  parts  of  the  several  tracks  which  would  lie 
on  different  leaves  of  the  surface  thus  constructed  are  indicated. 
The  initial  point  O  being  on  L0,  we  arrive  at  z  by  the  route 


FIG.  69. 


§  XXIII.]  THE  MODULUS  OF  A   SUM  425 

through  C  on  the  leaf  L\ ;  by  the  route  through  E  and  F,  we 
arrive  on  the  leaf  L2',  and,  by  the  route  through  D,  we  arrive 
on  the  leaf  L_I. 

The  Modulus  of  a  Sum. 

392.  The  sum  of  several  complex  quantities  a,  b,  . . . ,  I  is 
graphically  represented  by  the  straight  line  OL,  where  OA,  AB, 
.  . .  ,  KL  (equal  to  the  moduli  of  the  given  quantities)  are  laid 
off  successively  each  in  its  proper  direction,  forming  a  broken  line 
beginning  at  the  origin.  Therefore  the  modulus  of  the  sum,  which 
is  the  length  of  OL,  cannot  exceed  the  sum  of  the  moduli,  which 
is  the  length  of  the  broken  line  OBC  .  .  .  L.  It  is  in  fact  equal 
to  this  sum  only  when  the  arguments  of  all  the  terms  are  the 
same,  so  that  the  broken  line  becomes  a  straight  line. 

Applications  of  this  principle  in  the  discussion  of  infinite 
series  occur  in  the  following  articles.  We  here  notice  its  applica- 
tion to  the  value  of  a  definite  complex  integral.  Let 


where  ^  denotes  a  specified  track  of  integration  in  the  z-plane. 
While  z  describes  the  track  L,  w  describes  a  track  in  the  w-plane. 
Regarding  the  integral  as  the  sum  of  its  elements,  this  last  track 
is  the  limiting  form  of  the  broken  line  representing  the  summation 
of  the  elements,  hence  its  length  is  the  limit  of  the  sum  of  the 
moduli  of  the  elements.  If  5  is  the  length  of  arc  in  the  path  of 
z,  ds  is  the  modulus  of  dz,  and  if  /t  is  the  variable  modulus  of 

/(z),  fj.  ds  is  the  modulus  of  an  element;  thus      //  ds  is  the  length 

JL 

of  the  track  in  the  -zf-plane.  The  modulus  of  w  therefore  cannot 
exceed  this  integral.  Let  M  be  the  greatest  value  of  /*  for  points 


426  FUNCTIONS  OF   THE  COMPLEX   VARIABLE.    [Art.  392. 


on  the  track  L,  and  let  /  be  the  length  of  that  track;    it  follows 
a  fortiori  that  the  modulus  of  w  cannot  exceed  ML 

Power  Series  in  the  Complex   Variable. 

393.  A  series  of  complex  terms  is  convergent  when  the  point 
representing  the  sum  of  n  terms  tends  to  a  fixed  limiting  position 
as  n  increases  without  limit.  Among  non-convergent  series,  we 
may  distinguish  between  oscillating  series  for  which  this  point 
remains  at  a  finite  distance  (but  does  not  approach  a  limiting 
value),  and  properly  divergent  series  for  which  the  point  recedes 
indefinitely  from  the  origin. 

A  series  is  necessarily  convergent  when  the  moduli  of  its 
terms  form  a  convergent  series  (the  terms  of  which  are  of  course 
all  positive).  Such  a  series  is  said  to  be  absolutely  convergent. 

For  example  the  series 


D 


FIG.  70. 


is  absolutely  convergent 
when  z  is  a  complex  quan- 
tity whose  modulus  is  less 
than  unity.  In  Fig.  70  the 
sum  of  a  number  of  terms 
of  this  series  is  constructed. 
The  broken  line  OABC  .  .  . 
here  consists  of  links  whose 
lengths  form  a  decreasing 


geometrical  progression,  and  their  inclinations  to  the  axis  of 
reals  an  arithmetical  progression,  so  that  the  angles  at  A,  B, 
C,  etc.,  are  all  equal.  Such  a  polygon  can  be  inscribed  in  an 
equiangular  spiral,*  hence  the  pole  P  of  this  spiral  is  the 


*  See  Diff.  Calc.,  Art.  324.     If  successive  radii-vectores  of  an  equiangular 
spiral  make  equal  angles  at   the  pole,  they  are  in  geometrical  progression,  so 


§  XXIII.]  POWER  SERIES  IN  THE  COMPLEX  VARIABLE.          427 

limiting  position  of  the  nth  vertex  when  n  increases  without 
limit,  and  OP  represents  the  sum  of  the  series. 

394.  When  the  modulus  of  z  is  greater  than  unity,  the  nth 
vertex  of  the  broken  line  is  on  an  equiangular  spiral  but  recedes 
from  the  pole  indefinitely.  In  the  intermediate  case,  when  the 
mcdulus  is  unity,  the  polygon  is  plainly  one  inscribed  in  a  circle. 
We  can  only  say  of  the  nth  vertex  that  it  must  lie  somewhere  on 
this  circle;  therefore  the  sum  of  n  terms  does  not  tend  toward 
a  limit,  but  oscillates  about  a  mean  value. 

An  example  of  a  convergent  series  which  is  not  absolutely 
convergent  is  furnished  by  the  series 

Z  +  l22+j23+---+-Z*+-", 

ft 

when  the  modulus  of  z  is  unity.     The  series  of  moduli  is  the 

harmonic  series  i  +  \  +  J  H ,  which  we  have  seen  has  an  infinite 

sum  (Diff.  Calc.,  Art.  180).  But,  when  z  is  a  complex  unit, 
the  series  is  convergent;  the  broken  line,  though  of  infinite 
length,  is  wrapped  about  a  limiting  point.  So  also,  when  z=  —  i, 
we  have  a  series  of  alternately  positive  and  negative  terms,  and 
the  sum  converges  to  the  limit  log  2.* 

that  the  triangles  formed  by  joining  their  extremities  are  similar.  Hence  the 
chords  are  in  geometrical  progression  and  they  make  equal  angles  with  one 
another. 

*  The  peculiarity  of  the  absolutely  convergent  series  is  that  there  is  but  one 
limiting  value,  independent  of  the  order  of  the  terms,  whereas  in  the  other  case 
any  limiting  values  or  even  an  infinite  value  may  be  reached  as  the  result  of  a 
different  law  of  succession  of  the  terms.  For  example,  in  the  series  of  real 
terms  of  alternating  signs  considered  above,  the  sum  of  the  terms  of  either 
sign  is  infinite,  and  it  is  only  when  taken  alternately  that  their  sum  approaches 
the  limit  log  2.  A  series  of  this  kind  is  said  to  be  semi-convergent.  An  analogous 
case  in  infinite  products  is  considered  in  Art.  344,  see  foot-note. 


428  FUNCTIONS  OF   THE  COMPLEX   VARIABLE.     [Art.  395. 

Circle  and  Radius  of  Convergence. 
395.  If  the  power  series 

A+Bz+Cz2+Dz3+-- (i) 

is  convergent  for  a  given  value  Z,  it  will  be  convergent  for  any 
value  of  z  whose  modulus  is  less  than  that  of  Z. 

Let  R  be  the  modulus  of  Z  and  r<R  the  modulus  of  z,  then 

r     r2      r3 


is  a  convergent  geometrical  series  whose  sum  is  R/R  —  r.  Now, 
since  the  series 

A+BZ+CZ2+DZ*+--- (3) 

is  by  hypothesis  convergent,  its  terms  are  all  finite.  Let  M 
denote  the  greatest  modulus  of  any  term  of  the  series  (3).  The 
terms  of  the  series  (2)  are  the  ratios  of  the  moduli  of  correspond- 
ing terms  of  (i)  and  (3);  hence 

/       r     r^     r3  \~    MR 

\I+R+R2+R3+' 

exceeds  the  sum  of  the  moduli  of  the  series  (i).  It  follows  that 
these  moduli  form  a  convergent  series,  and  the  series  (i)  is  ab- 
solutely convergent  when  r<R. 

396.  A  power  series  may  be  convergent  for  all  values  of  z. 
If  it  is  not,  let  R  be  the  greatest  modulus  of  any  point  for  which 


§  XXIII.]  CIRCLE  AND  RADIUS  OF  CONVERGENCE.  429 

it  is  convergent,  then  it  will  be  convergent  for  every  point  whose 
modulus  is  less  than  R,  that  is  to  say,  for  every  point  within 
the  circle  whose  radius  is  R  and  centre  at  the  origin.  Again,  it 
will  be  divergent  for  every  point  outside  of  this  circle,  because 
R  is  by  hypothesis  the  greatest  modulus  for  which  it  is  convergent. 
This  circle  is  called  the  circle  of  convergence  for  the  series,  and  R 
is  called  the  radius  of  convergence. 

It  is  to  be  noticed  that  nothing  is  proved  as  to  the  points  on 
the  circumference  of  the  circle.  The  extremities  of  the  diameter 
lying  in  the  axis  of  reals  are  the  "limits  of  convergence"  for  power 
series  of  the  real  variable,  and  we  have  seen  that  such  a  series 
may  be  convergent  at  one  limit  and  divergent  at  the  other  (Diff. 
Calc.,  Art.  182). 

Taylor  s  Series. 

397.  The  equations  of  Art.  385  may  be  written  in  the  form 

i    f  <f>(z)dz 

q>(a)=  —  — , 

2^7tJc  z  —  a 

ju(n\      I   f    ^dz 

0'(a)=— r        ; T7>, 

2^7^:Jcz-a}2 


no  critical  point  of  the  function  <j>(z)  being  either  on  or  within 
the  contour  of  integration  C. 

These  equations  are  the  starting-point  for  many  investigations 
in  the  theory  of  functions.  They  enable  us,  for  example,  to  prove 
Taylor's  Theorem  for  the  complex  variable  and  to  determine  the 
radius  of  convergence  as  follows: 


430  FUNCTIONS  OF   THE  COMPLEX   VARIABLE.     [Art.  398.. 


398.  Let  the  contour  C  be  a  circle  with  radius  r  about  the 
point  a  as  centre,  Fig.  71,  and  let  k,  the 
modulus  of  h,  be  less  than  r,  so  that  the 
point  a+h  is  within  the  circle.  Since  the 
circle  is  a  contour  about  a+h  and  there  is 
no  critical  point  of  <£(z)  either  on  or  within 
its  circumference, 


FIG.  71. 

Now,  n  being  any  positive  integer, 
i  i  h  h" 


/  x 
u; 


h 


"+I 


z-a-h     z-a     (z-a}2  (z-a)**        (z-a)n+l(z-a-hy 
Substituting  in  equation  (i), 

0(<z +/£)  =  — r  —  +h\ \2~'"'"'"^*     7 vT+l~l~  U^i 

where 


R,= 


Hence 


h2 


We  have  now  to  show  that  the  remainder  /?„  has  zero  for 
its  limit  when  n  increases  without  limit.  For  this  purpose,  con- 
sider the  modulus  of  R  .  The  modulus  of  z— a  remains  constant 

n 

and  equal  to  r  while  z  moves  around  the  contour.     We 
therefore, 


mod.  R^  = 


mod 


.  f  - 

Jcz-a 


§  XXIII.]  TAYLOR'S  SERIES.  431 

The  integral  in  this  expression  is  finite  by  Art.  392  because  <£(z) 
is  by  hypothesis  finite  for  every  point  of  the  contour  and  z  —  a  +  h 
does  not  vanish.  Hence,  because  the  fraction  k/r  is  less  than 
unity,  the  modulus  of  Rn  decreases  without  limit  as  n  increases, 
that  is  to  say,  the  series  is  absolutely  convergent. 

Let  R  be  the  distance  from  a  of  the  nearest  critical  point  of 
<£(z) :  then  r  may  be  any  quantity  less  than  R  and  as  near  to  it 
as  we  please.  Therefore  R  is  the  radius  of  convergence. 

399.  The  result  of  putting  a  =  o,  and  writing  z  in  place  of  h, 
is  of  course  Maclaurin's  series 

/(Z)=/(0)+/'(0)Z+/"(0)^  +    ..-    +/1">(0)~.+  -", 

2  •  '  *  • 

in  which  f(z)  is  developed  into  a  series  involving  positive  integral 
powers  of  z.  Hence,  if  the  origin  is  not  itself  a  critical  point,  the 
function  can  be  developed  in  powers  of  z  within  a  circle  whose 
radius  R  is  the  least  modulus  of  a  critical  point. 

For  example,  in  the  case  of  the  function  tan-1z,  its  integral 
form,  Art.  386,  shows  that  the  limits  of  convergence  for  the 
corresponding  series  are  ±i,  because  unity  is  the  modulus  of 
each  of  the  poles  i  and  —i. 

A  one-valued  function  which,  like  0s,  is  infinite  only  for 
infinite  values  of  z  can  be  developed  into  a  series  convergent  for 
the  whole  plane. 

400.  We  can  now  show  that  a  one-valued  function  cannot 
fail  to   become   infinite,  for  at    least  one  value  of  z,  finite  or 
infinite.      For,  if  this  were  possible,  the   modulus  of  z  would 
have  some   finite  maximum  value  M.     Now  we  have,  for  auy 
point  a, 


432  FUNCTIONS  OF    THE  COMPLEX    VARIABLE.     [Art.  400. 

in  which,  since  <£(z)  has  by  hypothesis  neither  pole  nor  branch- 
point, the  contour  C  may  be  taken  as  a  circle  with  centre  a  and 
any  radius.  Putting,  then,  z=  a  +re*e,  whence  dz=  iret9dd,  we  have 


n\  f2*  dtz^ire*6  dd       n\    f 
a)  =  — 

2^xJ0  rn+1e(tt    °0     2arf*J 


The  modulus  of  the  integral  in  the  last  expression  is,  by  Art.  392, 
less  than  2nM;  hence  the  modulus  of  <f>("\a)  cannot  exceed 
n\  M/rn.  In  this  expression  r  can  be  made  as  large  as  we  choose, 
hence  <f>'(a)  <j>"(a)  etc.,  all  vanish,  and  equation  (2),  Art.  398, 
becomes  <j>(a+h)=<f>(a);  that  is  to  say,  <j>(z)  reduces  to  a  con- 
stant and  is  not  a  function  of  z. 

Since  this  applies  also  to  the  function  1/^(2),  it  follows  that 
^(2)  must  become  zero,  either  when  z  is  infinite  or  for  some 
finite  value  of  2.  In  particular,  if  <j>(z)  is  a  rational  integral 
function  we  have  thus  proved  the  fundamental  theorem  of 
Algebra  that  </>(2)=o  must  have  a  root. 

Examples  XXIII. 

1.  Putting  w=pe^=z2,  derive  the  polar  equations  of  the  images 
of  lines  parallel  to  the  axes  of  x  and  of  y.     Compare  Art.  371. 

2O2  2b2 

P  = r  >  P  = r- 

i+cos  ip         i  — cos  $ 

2.  Derive  conjugate  functions  from  w=z3,  and  thence  the  equations 
of  the  images  of  x=a  and  y=b. 

u=x3—T.xy2.       2ja3v2=(a3—u)( 

\J       S      *  I  \  /  \ 


3.  Derive  conjugate  functions  from  w=ee1  and  show  that  the 
net-work  of  lines  parallel  to  the  axes  in  the  z-plane  is  transformed 
into  concentric  circles  and  their  radii,  while  the  images  of  all  other 
straight  lines  are  equiangular  spirals. 

eF  cos  y,     (?  sin  y. 


§  XXIII.]  EXAMPLES. 


433 


4.  Show  that  u=a  and  v=b  give  for  the  inverse  of  a  given  function 
the  systems  of  curves  corresponding  to  parallels  to  the  axes.      Thence 
show  that,  for  the  function  2=  log  w,  these  systems  are 

x  =  log  secy+C,        #  =  log  cosec  ;y + C'. 
Verify  that  these  curves  cut  orthogonally. 

5.  Show  that,  for  the  function  square-root,  both  systems  of  curves 
are  rectangular  hyperbolas;  and,  for  the  cube-root,  each  consists  of 
cubics  having  three  fixed  asymptotes. 

6.  Show  that  the  polar  equations  of  the  images  of  x=a  and  y=b 
in  Ex.  2  are 

p=  a3  sec3  ^0,        p=  b3  cosec3  ^0. 

Trace  these  curves  for  a=i  and  b=i  and  verify  their  orthogonal 
intersection  at  (—2,  2).  Show  also  that,  at  the  other  two  points  of 
intersection,  these  curves  must  cut  each  other  at  angles  of  30°. 

Notice  that  the  image  of  x=  i  is  also  the  image  of  other  straight  lines 
in  the  z-plane. 

7.  Given  w=zn,  show  that  lines  parallel  to  the  axes  are  converted 
into  circles  when  n=  —  i;  into  cardioids  when  n=2\  and  into  lemnis- 
cates  when  n=  —  \. 

8.  If  w=  -j/z,  show  that,  when  z  describes  a  small  circle  about  a  given 
point  a,  the  two  branches  of  the  function  w  will  describe  the  separate 
branches  of  a  Cassinian  (Diff.  Calc.,  p.  313,  Ex.  29);  if  the  circle  passes 
through  the  origin,  the  ovals  merge  into  a  lemniscate;  and  if  it  encloses 
the  origin,  the  two  branches  of  the  function  describe  halves  of  a  con- 
tinuous oval. 

9.  Given  w=^/z,  assuming   the  initial  values  z0=i,  H>O=I,  draw 
a  track  of  z  which  will  make  w  the  real  cube-root  of  a  negative  quantity. 

10.  Putting  c=i,  b—  —  i  in  the  example  given  in  Art.  375,  so  that 

w=V(z2-i), 
show  that  the  systems  of  hyperbolas 

x2—y*=A        and        xy=B 


434        FUNCTIONS  OF  THE  COMPLEX   VARIABLE.  [Ex.  XXIII. 
form  an  orthogonal  net- work,  of  which  the  images  are 
u2—v2=A  —  i        an(j        uv=B, 

in  the  w-plane.  Show  by  construction  that  arcs  of  the  hyperbolas 
X2_y2—^  Xy=—^^  x2—y2=%,  xy=&  form  a  contour  by  which  z 
can  pass  from  the  value  z0=f+^,  around  the  branch-point  z=i, 
back  to  z0.  Then,  by  tracing  the  images  of  these  hyperbolas,  show  that 
iv,  starting  from  the  value  w0=i+fi,  will,  while  z  completes  the  con- 
tour, pass  from  iv0  to  —  w  in  accordance  with  Art.  375. 

11.  If7f=sinz,  we  have 

u=  sin  x  cosh  y,        v=  cos  x  sinh  y 

(Diff.  Calc.,  Art.  220);  show  that  the  orthogonal  net-work  in  the  to- 
plane  corresponding  to  lines  parallel  to  the  axes  in  the  z-plane  consists 
of  confocal  ellipses  and  hyperbolas. 

Note  that  for  the  inverse  function,  z=sin-1  w,  the  foci  are  the  branch- 
points, and  the  ellipse  furnishes  an  example  of  the  contour  about  both 
branch-points  equivalent  to  the  circuit  of  a  pole.  See  Art.  389. 

12.  Given  w=log  — r,  show  that  lines  parallel  to  the  axes  in  the 

%      0 

w-plane  correspond  to  two  systems  of  circles  in  the  z-plane.     If  the 
z-plane  be  cut  by  a  circular  arc  joining  a  and  b,  the  cut  plane  corre- 
sponds to  a  horizontal  strip  of  the  w-plane. 
Put  z—a^pie*^,  z—b=p2ei8*. 

13.  Putting  a=  —i  and  b=  i  in  example  12,  show  that  w—2i  cot-1z, 

w 

whence  z=  cot  — ..     Hence  show  how  to  construct  two  circular  arcs  in 
2^ 

the  z-plane  whose  intersection  determines  z=cot  (c+id). 

A  and  B  being  the  points  —  i  and  i,  one  passes  through  the  points 
whereas  is  cut  harmonically  in  the  ratio  e~zd;  the  other  is  a  segment 
on  AB  containing  the  angle  2c. 

14.  Prove  the  result  mentioned  in  the  foot-note  on  p.  415  by  in- 
tegration along  a  semicircular  track. 


1  6.  Find  the  value  of       -  —  ^~  -  —dz  when  the  contour  C  includes 


§  XXIII.]  EXAMPLES.  435 

15.  Show  geometrically  that,  when  z  describes  a  circle  of  radius  a 
about  the  origin,  the  value  of  dz/z  is  ids  /a;  and  thence  derive  the 
value  of  the  loop-integral. 

-  —  ^~  - 
Jc(a-a)(s- 

a  and  b  but  no  critical  point  of  <£(z);  and  thence  deduce  equation  (2), 
Art.  385. 

17.  Find  the  values  of  the  loop-integrals  of  dz/z2+i. 

±2 

1  8.  Find  the  values  of  the  loop-integrals  of  dz/z3—i. 


19-  Given  I  ~  .  ^>  find  the  values  of  the  loop-integrals  for  the  poles 

in  the  first  and  second  quadrants,  and  show  that  the  result  of  integrating 
in  the  positive  direction  about  these  two  poles  is  equivalent  to  the  recti- 
linear integral  from  —  oo  to  +  oo .  TT(I  —  i) 

M=    24/2    • 
Compare  Ex.  XX,  2. 

f       dz 

20.  Given  I- sr— r,  find  the  result  of  integration  in  a  contour 

J(l+Z3)|/Z 

including  the  three  poles  but  not  the  branch-point,  assuming  the  argu- 
ment of  z  to  lie  always  between  o  and  271.  |/T. 

21.  Construct   the   point  representing  the  generating  function  of 
the  series  represented  by  Fig.  70,  and  show  that  it  is  the  pole  of  the 
equiangular  spiral  mentioned  in  Art.  393.     Show  also  that,  when  the 
modulus  of  z  is  unity,  it  is  the  centre  of  the  circle  mentioned  in  Art.  394. 


INDEX. 


[The  numbers  refer  to  the  pages.] 


ABSOLUTE   convergence,  426,  427  foot- 
note 

Absolute  value,  or  modulus,  404 
Amsler's  planimeter,  211 
Analytical  functions,  409  foot-note 
Approximate   methods  of    quadrature, 

215  el  seq. 

Area  of  integration,  159,  161 
Areal  element,  or  lamina,  164 
Areas    described   by   moving    straight 
lines,  129,  206  et  seq. 

of  closed  circuits,  210 

plane,  3,  129  et  seq.,  172 

polar  element  of,  172 

polar  formulae,  134,  135,  136 
Argument,  404 
Arithmetical  mean,  or  average,  224 

BERNOULLI'S  series,  95 
Beta-function,  372 
Branch  of  a  function,  414 
Branch-point,  410,  413 
Buffon's  probability  problem,  269 

CARDIOID,  143  ex.  21,  183, 187  ex.  g,  188 

ex.  17,  190,  190,  207,  242  ex.  26 
Cassinian,  433  ex.  8 
Catenary,  32  ex.  77,  193  ex.  i,  203  ex.  2 
Cauchy's  general  and  principal  values, 

317,  3'9 
Centre  of  gravity,  234,  235 

of  position,  232 
Centroids,  235 

method  of  (in  mean  areas),  258 


Change  of    independent   variable,   50, 

119,  306 
Change   of  order  of  integration,    160, 

3°3 

Circle  of  convergence,  428 
Cissoid,  76  ex.  51,  136,  150,  151  ex.  3, 

242  ex.  17 

Closed  circuit,  area  of,  210 
Companion  to  cycloid,  13  ex.  *g 
Complex  values  of  constant  in  definite 

integrals,  310 
Complex  variable,  404  et  seq.,  426 

integration,  414 
Conchoid,  154  ex.  18,  187  ex.  9 
Conformal  representation,  407 
Conjugate  functions,  407 
Constant  of  integration,  2 
Continued  products,  374 
Continuous  variation,  106,  405 
Contour,  411,  416 
Contour  integral,  417,  420,  423 
Convergence,  426  et  seq. 
Corrected  integral,  106 
Cotes'  method  of  approximation,  218 
Critical  points,  416 
Current  variable,  296,  415 
Curtate  cycloid,  142  exs.  13,  14 
Curves  of  probability,  276,  280,  281 
Cusp,  190 

Cuts  in  the  z-plane,  414,  424 
Cycloid,  48  ex.  48,  152   exs.  8,  9,  194 

exs.  6,  7,  204  exs.  5,  6,  7,  208,  242 

ex.  19 

Cylindrical  coordinates,  176 
element  of  volume,  150,  239 

437 


INDEX. 


DEFINITE  integrals,  4,  106,  296  et  seq. 

differentiation  of,  298 
Definite  integrals,  integration  of,  300 

obtained  by  expansion,  329  el  seq. 
Density,  variable,  166,  237,  242  ex.  20 
Derivative,  complex  values  of,  406 
Development  of  an  integral  in  series,  94, 

329,  330 
Differential  of  an  area,  3,  130,  134,  137 

of  a  volume,  147, 150 
Direct  integration,  14 
Discontinuity,  350,  361 
Discontinuous  functions,  329,  345 
Divergent  series,  406 
Double  curvature,  curves  of,  191 
Double  integrals,  155  et  seq.,  174,  303 

transformation  of,  174,  307 

ELEMENT  of  an  integral,  123,  156,  163 

of  area,  172 

of  volume,  176,  178,  181 
Elementary  theorems,  6 
Ellipse,  48  ex.  27,  140  ex.  7,  154  exs. 

19,  20 

Epicycloid,  196  ex.  14 
Equiangular  spiral,  197  ex.  17,  426 
Eulerian  integrals,  372  et  seq.,  381 
Euler's  constant,  390 

FACILITY  of  errors,  law  of,  281 
Favorable  and  unfavorable  cases,  266 
Folium,  137 

Fonctions  monogene,  409  foot-note 
Formulae  of  integration,  8,  124 
Formulae  of  reduction: 

algebraic  forms,  90  et  seq.,  103  et  seq. 

definite  integrals,  116  et  seq. 

indefinite  integrals,  81  et  seq. 

trigonometric  forms,  82  et  seq.,  89, 102 
Four-cusped  hypocycloid,  242  ex.  18 
Fourier's  series,  342  et  seq. 

double  integral  theorem,  364 
Frullani's  integral,  325 
Functions  of  the  complex  variable,  404 

et  seq. 
Fundamental  integrals,  8,  124 

GAMMA  functions,  377  et  seq. 
graph  of,  378 
table  of,  401 

r,  39°.  392 


Gauss'  II -function,  374 

theorem  in  /"-functions,  396 

Graph  of  an  integral,  99  et  seq. 
of  the  /"-function,  378 

Graph  of  multiple-angle  series,  344  et 

se1->  353 
Gyration,  radius  of,  237 

HARMONIC  series,  389 

Hoi  ditch's  theorem,  215  ex.  n 

Hyperbola,  76  ex.  50,  142  ex.  15,  170  ex. 

12,  205  ex.  10 
Hyperbolic  (Naperian)  logarithms,   13 

ex.  25 

Hyperbolic  sine  and  cosine,  146  ex.  32 
Hyperbolic  spiral,  143  ex.  23 
Hypocycloid,  242  ex.  18 

IMAGE,  405 
Increment,  107 
Indefinite  integral,  5,  ic6 
Inertia,  moment  of,  237 
Infinite  values  of  the  element,  316 
Infinite  values  of  the  limits,  no,  318 
Initial  value,  4,  106,  410,  415 
Instantaneous  centre,  206  joot-note 
Integral,  2,  105,  123  joot-note 

definite,  4,  53,  107,  296  et  seq. 

double,  155  et  seq.,  303 

indefinite,  5,  106 

regarded  as  limit  of  a  sum,  121 

triple,  163,  1 66 
Integrand,  415 
Integration  about  a  pole,  417 

around  a  branch-point,  420 

by  expansion,  94,  329 

by  parts,  77  et  seq. 

by  transformation,  33,  50 

direct,  14 

of  forms  containing  radica's,  59,  421 

of  rational  fractions,  15  et  seq. 

of  trigonometric  forms,  33  et  seq. 

over  an  area,  159,  161 

over  a  volume,  165 

under  the  integral  sign,  299 
Intermediate  values,  107,  415  foot-note 

LEMNISCATE,  135,  179,  186  ex.  7 
Limacon,  144  exs.  28,  29,  214  ex.  5 
Limit  of  summation,  121 
Limits  of  an  integral,  5,  105 


INDEX. 


439 


LimiiS    of   application   of   a    Fourier's 

series,  342 ,  360,  362 
of  a  double  integral,  156,  161 
of  a  transformed  integral,  53 
of  a  triple  integral,  164 

Linear  element,  164 

Local  probability,  269  et  seq. 

Log  r(i+  *),  389 

Logarithmic  derivative  of  F(x),  392 

Loop,  412,  416 

Loop-integral,  418 

Loxodromic  curve,  192 

MAGNIFICATION  (in  mapping),  407 
Mapping,  or  conformal  representation, 

4°5.  4°7 

Mean  areas,  256,  258 
Mean  distances:  • 

between  variable  points,  246,  248 

from  a  fixed  point,  243 

from  a  plane,  232 
Mean  ordinates,  218  foot-note,  226 
Mean  squared  distances: 

from  a  plane,  237 

from  an  axis,  238 

Mean  values,  224  foot-note,  227,  230 
Modulus,  404,  425 
Moment,  statical,  235 

of  inertia,  237,  238 

Multiple -angle  series,  332,  334,  341  el 
seq. 

differentiation  of,  353 

integration  of,  355 
Multiple -valued  functions,  412 
Multiple-valued  integrals,  112,  115 

NAPERIAN  logarithms,  13  ex.  25 
"  Number   of  cases"  of  a  continuous 
variable,  228,  230 

ORDER  of  integration,  160,  173,  304 
Oscillating  series,  426 

PARABOLA,  12  ex.  22,  132,  140  ex.  6, 
153  ex.  13,  193  ex.  2,  195  ex.  10, 
241  exs.  14,  16,  242  ex.  24 

Parabola  of  wth  degree,  12  ex.  24 

Parabo'oid,  147,  203  ex.  i,  242  ex.  25 

Partial  fractions,  15  et  seq. 
formulae  for  numerator,  22,  23 

Parts,  integration  by,  77  et  seq. 


Periodic  functions,  342,  420 

Plane  areas,  3,  129  et  seq.,  172 

Planimeter,  211 

Polar  coordinates  in  space,  179 

Polar  element  of  area,  134,  172 

Polar  element  of  area,  transformed,  136 

Pole,  417 

Power  series  in  the   complex  variable, 

426 

Primary  values,  113,  115,351,  412,  420 
Principal  values  of  an  integral,  317,  319 
Probability,  266 

curves  of,  276,  280,  281 
Probable  value,  278 

QUADRATURE,  approximate,  215 

comparison  of  rules,  220  foot-note,  223 

ex.  5  and  foot-note 

Quasi-periodic  functions,  335,  352,  353 
Quick-return  motion,  352 

RADICALS,  integrals  involving,  59  et  seq. 

421 

Radius  of  convergence,  428 
Radius  of  gyration,  237 
Random  direction,  231 

lines,  287 

parts,  250  et  seq. 

points,  230,  256 

Rational  fractions,  15  et  seq.,  320 
Rational  integral  function,  existence  of 

a  root,  432 

Rectification  of  curves,  189  et  seq. 
Rectilinear  track,  412,  420,  423 
Reducible  tracks,  411 
Riemann's  surface,  414,  424 

SEMI -CONVERGENT  series,  427 
Semi-cubical  parabola,  241  ex.  15 
Series  in  powers  of  the  complex  vari-. 

able,  426 
in    sines    and    cosines   of    multiple 

angles,  332,  334,  341 
Maclaurin's,  431  . 
Taylor's,  96,  430 
Simpson's  rules,  217,  220  foot-note 
Solids  of  revolution,  178 
Sphere,  170  exs.  n,  12,  187  exs.  u,  12, 

241  ex.  10 
Spherical  coordinates,  147,  150,  181 


440 


INDEX. 


Spheroid,  203  exs.  2,  4 

Spiral  of  Archimedes,  143  ex.  19,  185 

ex.  i 

Stationary  point,  190 
Stereographic  projection,  407  fool-note 
Strophoid,  49  ex.  50,  76  ex.  52 
Summation,  limit  of,  121 
Surfaces  of  revolution,  198 
in  general,  199 

TABLES:  /"-functions,  401 
I    values  of  5n  =  5»—  i,  403 
Taylor's  theorem,  96,  430 
Track  of  complex  variable,  410 

of  integration,  415  , 

rectilinear,  412,  423 

reduced,  411 
Tractrix,  153  ex.  14,  194  ex.  5,  204  ex. 

8,  208 
Transformation : 

of  definite  integrals,  119,  306 


Transformation : 

of  double  integrals,  174 

of  triple  integrals,  166 
Triple  integrals,  163 
Two-valued  functions,  410 

ULTIMATE  element,  164,  173 
Uniform    distribution    of  values  of    a 
function,  228,  232 

VARIABLE  density,  166 
Viviani's  problem,  203 
Volume,  integration  over  given,  165, 

167 
Volumes,  147  et  seq.,  159,  168,  178,  182 

WEDDLE'S  rule,  219  foot-note,  223  foot- 
note 

Weighted  mean,  225,  234 
Witch,  32  ex.  78,  130,  152  ex.  4 
Woolley's  rule,  221 


495$  .  5 


Johnson  -  3!  treatise  on  the  integral 


calculus  founded  on  the  method 


QK 


A    000  191  050    4 


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